Draft

STRANGE ATTRACTORS

Creating Patterns in Chaos

Julien C. Sprott

The University of Wisconsin

Madison Wisconsin

Copyright (c) 1993 by Julien C. Sprott

Contents

WHY THIS BOOK IS FOR YOU

CHAPTER 1: ORDER AND CHAOS

1.1 Predictability and Uncertainty

1.2 Bucks and Bugs

1.3 The Butterfly Effect

1.4 The Computer Artist

CHAPTER 2: WIGGLY LINES

2.1 More Knobs to Twiddle

2.2 Randomness and Pseudo-randomness

2.3 What's in a Name?

2.4 The Computer Search

2.5 Wiggles on Wiggles

2.6 Making Music

CHAPTER 3: PIECES OF PLANES

3.1 Quadratic Maps in Two Dimensions

3.2 The Butterfly Effect Revisited

3.3 Searching the Plane

3.4 The Fractal Dimension

3.5 Higher Order Disorder

3.6 Strange Attractor Planets

3.7 Designer Plaids

3.8 Strange Attractors that Don't

3.9 A New Dimension in Sound

CHAPTER 4: ATTRACTORS OF DEPTH

4.1 Projections

4.2 Shadows

4.3 Bands

4.4 Colors

4.5 Characters

4.6 Anaglyphs

4.7 Stereo Pairs

4.8 Slices

CHAPTER 5: THE FOURTH DIMENSION

5.1 Hyperspace

5.2 Projections

5.3 Other Display Techniques

5.4 Writing on the Wall

5.5 Murals and Movies

5.6 Search and Destroy

CHAPTER 6: FIELDS AND FLOWS

6.1 Beam Me up Scotty!

6.2 Professor Lorenz and Dr. Rössler

6.3 Finite Differences

6.4 Flows in Four Dimensions

6.5 Strange Attractors that Aren't

6.6 Doughnuts and Coffee Cups

CHAPTER 7: FURTHER FASCINATING FUNCTIONS

7.1 Steps and Tents

7.2 ANDs and ORs

7.3 Roots and Powers

7.4 Sines and Cosines

7.5 Webs and Wreaths

7.6 Swings and Springs

7.7 Roll Your Own

CHAPTER 8: EPILOG

8.1 How Common is Chaos?

8.2 But is it Art?

8.3 Can Computers Critique Art?

8.4 What's Left to Do?

8.5 What Good is it?

APPENDIXES

A. Annotated Bibliography

B. BASIC Program Listing

C. Other Computers and BASIC Versions

D. C Program Listing

E. Summary of Equations

F. Dictionaries of Strange Attractors

INDEX

Acknowledgments

I am indebted to Professor George Rowlands of the University of Warwick for introducing me to chaos and fractals and for countless stimulating discussions.

Professor Edward Pope of the University of Wisconsin - Madison Art Department assured me that these patterns have artistic appeal and suggested ways to display them.

Dr. Clifford Pickover of the IBM Watson Research Center the guru of computer visualization provided encouragement and suggestions during the early development of the ideas on which this book is based.

I would also like to thank Ray Valdes for carefully reading the manuscript and providing numerous helpful suggestions.

Finally I am grateful to scores of individuals who have critically viewed my attractors and who collectively raised my artistic consciousness to the point where this book could become a reality.

Why This Book Is For You

Art and science sometimes appear in juxtaposition one aesthetic the other analytical. This book bridges the two cultures. I have written it for the artist who is willing to devote a modicum of effort to understanding the mathematical world of the scientist and for the scientist who often overlooks the beauty that lurks just beneath even the simplest equations.

If you are neither artist nor scientist but own a personal computer for which you would like to find an exciting new use this book is also for you. Fractals generated by computer represent a new art form that anyone can appreciate and appropriate. You don't have to know mathematics beyond elementary algebra and you don't have to be an expert programmer. This book explains a simple new technique for generating a class of fractals called strange attractors. Unlike other books about fractals that teach you to reproduce well-known patterns this one will let you produce your own unlimited variety of displays and musical sounds with a single program. Almost none of the patterns you produce will ever have been seen before.

To get the most out of this book you will need a personal computer though it need not be a fancy one. It should have a monitor capable of displaying graphics preferably in color. Some knowledge of BASIC is useful although you can just type in the listings even if you don't understand them completely. For those of you who are C programmers I have provided an appendix with an equivalent version in C. You may find the exercises in this book an enjoyable way to hone your programming skills. As you progress through the book you will gradually develop a very sophisticated computer program. Each step is relatively simple and brings exciting new things to see and explore. Alternately you can use the accompanying disk immediately to begin making your own collection of strange attractors.

It is my hope that this book will instill in the artist a greater appreciation of science and in the scientist a greater appreciation of art and that it will bring enjoyment and satisfaction to computer enthusiasts both new and seasoned.

CHAPTER 1

Order and Chaos

This chapter lays the groundwork for everything that follows. Nearly all the essential ideas mathematical techniques and programming tools are developed here. If you master the material in this chapter the rest of the book should be smooth sailing.

1.1 Predictability and Uncertainty

The essence of science is predictability. Halley's comet will return to the vicinity of the earth in the year 2061. Not only can astronomers predict the very minute when the next solar eclipse will occur but also the best vantage point on the Earth from which to view it. Scientific theories stand or fall according to whether their predictions agree with detailed quantitative observation. Such successes are possible because most of the basic laws of nature are deterministic which means they allow us to determine exactly what will happen next from a knowledge the present conditions.

However if nature is deterministic there is no room for free will. Even human behavior would be predetermined by the arrangements of the molecules that make up our brains. Every cloud that forms or flower that grows would be a direct and inevitable result of processes set into motion eons ago and over which there is no possibility for exercising control. Perfect predictability is dull and uninteresting. Such is the philosophical dilemma that often separates the arts from the sciences.

One possible resolution was advanced in the early decades of the twentieth century when it was discovered that the quantum mechanical laws that govern the behavior of atoms and their constituents are apparently probabilistic which means they allow us to predict only the probability that something will happen. Quantum mechanics has been extremely successful in explaining the sub-microscopic world but it was never fully embraced by some including Albert Einstein who until his dying day insisted that he did not believe that God plays dice with the Universe.

Science has since the 1970's been undergoing an intellectual revolution that may be as significant as the development of quantum mechanics. It is now widely understood that deterministic is not the same as predictable. An example is the weather. The weather is governed by the atmosphere and the atmosphere obeys deterministic physical laws. However long-term weather predictions have improved very little as a result of careful detailed observations and the unleashing of vast computer resources.

The reason is that the weather exhibits extreme sensitivity to initial conditions. A tiny change in today's weather (the initial conditions) causes a larger change in tomorrow's weather and an even larger change in the next day's weather. This sensitivity to initial conditions has been dubbed the butterfly effect since it is hypothetically possible for a butterfly flapping its wings in Brazil to set off tornadoes in Texas. Since we can never know the initial conditions with perfect precision long-term prediction is impossible even when the physical laws are deterministic and exactly known. It has been shown that the predictability horizon in weather forecasting cannot be more than two or three weeks.

Unpredictable behavior of deterministic systems has been called chaos and it has captured the imagination of the scientist and non-scientist alike. The word "chaos" was introduced by Tien-Yien Li and James A. Yorke in a 1975 paper entitled "Period Three Implies Chaos." The term "strange attractors " from which this book takes its title first appeared in print in a 1971 paper entitled "On the Nature of Turbulence " by David Ruelle and Floris Takens. Some people prefer the term "chaotic attractor " since what seemed strange when first discovered in 1963 is now largely understood.

It's not hard to imagine that if a system is complicated (many springs and wheels and so forth) and hence governed by complicated mathematical equations that its behavior might be complicated and unpredictable. What has come as a surprise to most scientists is that even very simple systems described by simple equations can have chaotic solutions. However everything is not chaotic. After all we can make accurate predictions of eclipses and many other things.

An even more curious fact is that the same system can behave either predictably or chaotically depending on small changes in a single term of the equations that describe the system. For this reason chaos theory holds promise for explaining many natural processes. A stream of water for example exhibits smooth (laminar) flow when moving slowly and irregular (turbulent) flow when moving more rapidly. The transition between the two can be very abrupt. If two sticks are dropped side-by-side into a stream with laminar flow they will stay close together but if they are dropped into a turbulent stream they quickly separate.

Chaotic processes are not random; they follow rules but even simple rules can produce extreme complexity. This blend of simplicity and unpredictability also occurs in music and art. Music consisting of random notes or of an endless repetition of the same sequence of notes would be either disastrously discordant or unbearably boring. Art produced by throwing paint at a canvas from a distance or by endlessly replicating a pattern like a piece of wallpaper is similarly unlikely to have aesthetic appeal. Nature is full of visual objects such as clouds and trees and mountains as well as sounds like the cacophony of excited birds that have both structure and variety. The mathematics of chaos provides the tools for creating and describing such objects and sounds.

Chaos theory reconciles our intuitive sense of free will with the deterministic laws of nature. However it has an even deeper philosophical ramification. Not only do we have freedom to control our actions but the sensitivity to initial conditions implies that even our smallest act can drastically alter the course of history for better or for worse. Like the butterfly flapping its wings the results of our behavior are amplified with each day that passes eventually producing a completely different world than would have existed in our absence!

1.2 Bucks and Bugs

Enough philosophizing--it's time to look at a specific example. This example will require some mathematics but the equations are not difficult. The ideas and terminology are important for understanding what is to follow.

Suppose you have some money in a bank account that provides interest compounded yearly and that you don't make any deposits or withdrawals. Let's let X represent the amount of money in your account. When the time comes for the bank to credit your interest its computer does so by multiplying X by some number. If the interest rate were 10% the number would be 1.1 and your new balance would be 1.1 X. If your balance in the n'th year is Xn (where n is 1 after the first year 2 after the second and so forth) your balance in the year n +1 is

Xn +1 = R Xn (Eq. 1A)

where R is equal to 1.0 plus your interest rate. (R is 1.1 in this example.)

You probably know that such compounding leads to exponential growth. In terms of the initial amount X0 the amount in your account after n years is

Xn = X0Rn (Eq. 1B)

After 50 years at 10% yearly interest you will have $117.39 for every dollar you initially had invested. The bank can afford to do this only because of inflation and because money is loaned at an even higher interest rate.

Equation 1A is applicable to more than compound interest. It's how many of us have our salaries determined. It also describes population growth. Imagine some species of bug that lives for a season lays its eggs and then dies (thus avoiding the confusion of overlapping generations). The next year the eggs hatch and the number of bugs is some constant R times the number in the previous year. If R is less than 1 the bugs die out over a number of years and if R is greater than 1 their number grows exponentially.

You also know that exponential growth cannot go on forever whether it be bucks in the bank or people on the planet. Eventually something happens such as the depletion of resources and the growth slows and perhaps even reverses. Mass starvation disease crime and war are some of nature's mechanisms for limiting unbridled population growth. Thus we need to modify the above equation in some way if it is to model more closely growth patterns in nature.

Perhaps the simplest modification is to multiply the right-hand side of Equation 1A by a term such as (1 - X) that is equal to 1 if X is small (much less than 1) but which is less than 1 as X increases. Since the growth slows to zero and reverses as X approaches 1 we must think of X = 1 as representing some large number of dollars or bugs (say a million or a billion); otherwise we would never get very far! And so our modified equation called the logistic equation is

Xn +1 = R Xn (1 - Xn) (Eq. 1C)

Now you're going to get your first homework assignment. Take your pocket calculator and start with a small value of X say 0.1. To reduce the amount of work you have to do use a fairly large value of R say 2 corresponding to a doubling every year. Run X through Equation 1C a few times and see what happens. This process is called iteration and the successive values are called iterates. If you did it right you should see that X grows rapidly for the first couple of steps and then it levels off at a value of 0.5. The first few values should be approximately 0.1 0.18 0.2952 0.4161 0.4859 0.4996 and 0.5. Compare your results with the unbounded growth of Equation 1A.

You might have predicted the above result if you had thought to set Xn+1 equal to Xn in Equation 1C and solved for Xn. This value is called a fixed-point solution of the equation because if X ever has that value it will remain fixed there forever. It is also sometimes called a point attractor because every initial value of X between 0 and 1 is attracted to the fixed point upon repeated iteration of Equation 1C. Try initial values of X = 0.2 and X = 0.8. A fixed point is also sometimes called a critical point a singular point or a singularity.

If you're curious you might wonder what happens if you start with a value of X less than 0 such as -0.1 or greater than 1 such as 1.1. You should verify that the iterates are negative and that they get larger and larger eventually approaching minus infinity. We say that the solution is unbounded and that it attracts to infinity. Thus the values of X = 0 and X = 1 are like a watershed. Between these values the solution is bounded and outside these values it is unbounded.

The region between X = 0 and X = 1 is called a basin of attraction since it resembles a bathroom basin in which drops of water find their way to the drain from wherever they start. X = 0 is also a fixed point but it is unstable since values either slightly above or slightly below zero move away from zero. Such an unstable fixed point is sometimes called a repellor. Chaos results when two or more repellors are present; the iterates then bounce back and forth like a baseball runner caught in a squeeze play.

Equations that exhibit chaos have solutions that are unstable but bounded; the solution never settles down to a fixed value or even to a repeating pattern but neither does it move off to infinity. Sometimes we say that such equations are linearly unstable but nonlinearly stable. Small perturbations to the system grow but the growth ceases when the nonlinear terms become important as eventually they must. Another way to say it is that the fixed points are locally unstable but the system is globally stable. In such a case initial conditions are drawn to a special type of attractor called a strange attractor which is not a point or even a finite set of points but rather a complicated geometrical object whose properties constitute the subject of this book.

See what happens if you substitute X = 0 or X = 1 into the logistic equation. As a check on your calculations or in case you didn't do your homework Table 1.1 shows the successive iterates of X for each of the cases we have discussed.

Table 1.1 Iterates of the logistic equation for various initial values of X with R=2

An equation such as the logistic equation that predicts the next value of a quantity from the previous value is called an iterated map because it is like a road map in which each point on the earth is mapped to a corresponding point on a piece of paper. The logistic equation is a one-dimensional map because the various X-values can be thought of as lying along a straight line that stretches from minus infinity to plus infinity. Each iteration of the map moves every point along the line to a new position on the line. For the example above with R = 2 all the points between X = 0 and X = 1 walk toward X = 0.5 where they stop and remain. Other points run faster and faster toward the end of the line that stretches to minus infinity.

The logistic equation is an example of a quadratic iterated map so called because if you multiply out the right-hand side of Equation 1C it has not only a linear term RXn but also a quadratic (squared) term -RXn2. Quadratic maps are noninvertable because you can find Xn+1 from Xn but you can't go backwards because there are two values of Xn that produce the same Xn+1 and there is no way of knowing from which it came. For example Table 1.1 shows that X0 = 0.2 and X0 = 0.8 both produce X1 = 0.32. These are the two roots of the quadratic equation that you get if you try to solve for Xn in Equation 1C in terms of Xn+1.

The graph of Xn+1 versus Xn is a curve called a parabola. Because a parabola is not a straight line the map is said to be nonlinear. Chaos and strange attractors require a nonlinearity. The interesting and surprising behavior of nonlinear iterated maps is the basis for much of this book.

The first surprising result occurs if you iterate Equation 1C with R = 3.2 and an initial value of X in the range of 0 to 1. After a few iterations the solution will alternate between two values of approximately 0.5130 and 0.7995. This is called a period-2 limit cycle. Like the fixed point the limit cycle is another type of simple attractor. It is sometimes called a periodic or cyclic attractor.

It's not hard to see how cyclic behavior might arise in nature. If the population of beetles grows too large they will deplete the plants on whom they depend for food. With too few plants the beetles die out allowing the number of plants to recover leading to the next cycle of beetle growth and so forth.

Increase R a bit more to 3.5 and repeat the calculation. The result is a period-4 limit cycle with four values of approximately 0.5009 0.8750 0.3828 and 0.8269. If you keep increasing R by ever smaller amounts the period of the limit cycle will double repeatedly finally reaching chaotic behavior (an infinite period) at about R = 3.5699456. This value is sometimes called the Feigenbaum point after Mitchell J. Feigenbaum who discovered many of the interesting properties of one-dimensional maps.

When chaos occurs the successive iterates fluctuate in an apparently random and irreproducible manner. The chaotic behavior persists up to R = 4 except for an infinite number of small periodic windows. For R greater than 4 the solution is unbounded and the iterates attract rapidly to minus infinity.

The behavior described above can be summarized in a bifurcation diagram as shown in Figure 1-1 in which the limiting iterated values of the logistic equation after discarding the first few hundred iterates are plotted for a range of R from 2 to 4. This plot is called the Feigenbaum diagram and it resembles a tree on its side ("Feigenbaum " appropriately but coincidentally is German for "fig tree"). You see the fixed-point solution for R less than 3 the period-doubling route to chaos and the periodic windows at large R. The chaotic regions toward the right side of the figure are characterized by values of X that span a wide range and eventually fill the region densely with points.

X versus R

Figure 1-1. Bifurcation Diagram for the Logistic Equation Xn+1 = RXn (1 - Xn)

Each period doubling is called a bifurcation since a single solution splits into a pair of solutions. These splittings are called pitchfork bifurcations for obvious reasons. Note the period-3 window at about R = 3.84. The period-3 region begins abruptly when R is increased slightly from within the chaotic region to its left in what is called a tangent or saddle-node bifurcation. Careful inspection of the period-3 window shows that it also undergoes a period-doubling sequence at about R = 3.85. Solutions with every period can be found somewhere between R = 3 and R = 4.

Successive period doublings occur with ever increasing rapidity as one moves from left to right in Figure 1-1. The ratio of the width of each region to the width of the previous region approaches a constant equal to about 4.6692 called the Feigenbaum number. Even more remarkable is that this number arises in many different chaotic systems in nature as well as in the solutions of equations. The universality of the Feigenbaum number in chaos is reminiscent of the ubiquity of the number ¹ in Euclidean geometry.

With R = 4 the solutions occupy the entire interval from X = 0 to X = 1. Eventually X will take on a value arbitrarily close to any point in that interval (a characteristic called topological transitivity). Curiously however there are infinitely many initial values of X that don't lead to a chaotic solution even for R = 4. For example X0 = 0.5 and X0 = 0.75 lead to unstable fixed points while X0 = 0.345491... and X0 = 0.904508... produce an unstable period-2 limit cycle. By unstable we mean that if the initial values are wrong by even the slightest amount successive iterates will wander ever farther away.

Even through there are infinitely many non-chaotic initial values between zero and one the chance that you will find one by randomly guessing is negligible. For every such value there are infinitely many others that produce chaos. Such a seemingly paradoxical entity is an example of a Cantor set named after the nineteenth-century German mathematician Georg Cantor who is often credited with developing a mathematically rigorous concept of infinity.

A Cantor set contains infinitely many members (in fact uncountably infinitely many) but its members represent a zero fraction of the total! For example infinitely many points are required to cover completely the circumference of a circle but this number of points doesn't even begin to cover its interior. Such a collection (or set) of points although infinite in number is said to comprise a set of measure zero because the points fill a negligible portion of the plane. An attractor is a set of measure zero but its basin of attraction has a non-zero measure.

Few people would have guessed that such complexity could arise from such underlying simplicity. Furthermore the logistic equation is only the simplest of an endless variety of equations that can exhibit chaos. It is this dichotomy of simplicity and complexity that makes chaos beautiful to the mathematician and artist alike. In the bifurcation diagram of the logistic equation we have something with aesthetic appeal and it came from a simple quadratic equation!

1.3 The Butterfly Effect

If our goal is to seek chaotic behavior in the solution of equations we need a simple way to test for chaos. For this purpose we use the fact that chaotic processes exhibit extreme sensitivity to initial conditions in contrast to regular processes where different starting points usually converge to the same sequence of points on a simple attractor.

Suppose we iterate the logistic equation with two initial values of X that differ by only a tiny amount. Think of these values as representing two states of the atmosphere that differ only by the flapping of the wings of a butterfly. If successive iterates are attracted to a fixed point as they are for R = 2 the difference between the two solutions must get smaller and smaller as the fixed point is approached. A similar thing happens for a limit cycle. The difference between the two solutions will on average decrease exponentially.

If the solution is chaotic as is the logistic equation for R = 4 the successive iterates for the two cases will initially on average get farther apart; the difference usually grows exponentially. If the difference doubles on average with every iteration we say the Lyapunov exponent is 1. If it is reduced by half we say the Lyapunov exponent is -1. The name comes from the late nineteenth-century Russian mathematician Aleksandr M. Lyapunov (sometimes transliterated Liapunov or Ljapunov).

You can think of the Lyapunov exponent as the power of 2 by which the difference between two nearly equal X-values changes on average for each iteration. Thus the difference between the values changes by an average of 2L for each iteration. If L is negative the solutions approach one another; if L is positive we have sensitivity to initial conditions and hence chaos.

One way to detect chaos is to iterate the equation with two nearly equal initial values and see if after many iterations the values are closer together or farther apart. Another way is to make use of a principle of calculus that says that the difference in the solutions after one iteration divided by the difference before the iteration provided the difference is small is equal to the derivative of the equation for the map which for the logistic equation is

DXn+1 / DXn = R(1 - 2Xn) (Eq. 1D)

where DX is the difference between the two values of X. In Equation 1D ÆXn is the difference in the X values after n iterations and ÆXn+1 is the difference after n+1 iterations.

Since DX increases by the factor on the right of Equation 1D for each iteration the proper way to calculate the average is to start with a value of 1 and multiply it repeatedly by the right-hand side of Equation 1D at each iteration then divide the result by the number of iterations and finally take the logarithm to the base 2 of the absolute value of the result to get the Lyapunov exponent. If you prefer an equation the above description is equivalent to

L = S log2 |R (1 - 2Xn)| / N (Eq. 1E)

where the vertical bars mean that you are to disregard the sign of the quantity inside and S means to sum the quantity to its right from a value of n = 1 to a value of n = N where N is some large number. The larger the value of N the more accurate is the estimate of L.

Suppose you knew the value of X to within 0.01 for an iterated map with L = 1. After one iteration the uncertainty would be about 0.02 and after two iterations the uncertainty would be about 0.04 and so forth. After about seven iterations the error would exceed 1 and your prediction would be totally worthless. If the X-values are expressed as binary numbers each iteration would result in throwing away the right-most binary digit (bit). Thus the units of L are bits per iteration. The Lyapunov exponent is the rate at which information is lost when a map is iterated.

It is as if a succession of cartographers each copied maps from one another but every time one was copied it was only half as accurate as the previous one. If the original map were accurate to 1% the next copy would be accurate to 2% and the seventh generation copy would bear no relation to the original. If the Lyapunov exponent were -1 one bit of information would be gained at each iteration. Even a completely unknown initial condition would eventually be perfectly accurate as it approached the known fixed point or limit cycle. Unfortunately negative Lyapunov exponents are not the rule in cartography; otherwise all our maps would be self-correcting!

Figure 1-2 shows the Lyapunov exponent for the logistic equation using values of R from 2 to 4. The Lyapunov exponent is 1.0 at R = 4 since that value causes the interval of X from 0 to 1 to be mapped backed onto itself with a single fold at X = 0.5. Thus information is lost at a rate of 1 bit per iteration since each iterate has two possible predecessors. You can also see some of the periodic windows where L dips below zero toward the right edge of the plot. Also note that L is zero wherever a bifurcation occurs. At these points the solution is fraught with indecision over which branch to take and the initial uncertainty persists forever neither increasing nor decreasing.

L versus R

Figure 1-2. Lyapunov Exponent for the Logistic Equation

1.4 The Computer Artist

By now you have probably surmised that the operations described above are best carried out by a computer. The equations are simple but they must be applied repeatedly. This is precisely the kind of task at which computers excel.

There are dozens of computer types and programming languages to choose from. Currently the most popular computers are those based on the IBM PC running the MS-DOS or IBM-DOS operating system (hereafter simply called DOS). The most widely available programming language is BASIC (Beginner's All-purpose Symbolic Instruction Code) which usually comes bundled with the operating system software included with the computer. A version of BASIC called QBASIC has been included with DOS since version 5.0. BASIC may not be the most advanced computer language but it is one of the easiest to learn and to use its commands are close to ordinary English and it is more than adequate for our purposes. Furthermore modern versions of BASIC compare favorably with the best of the other languages.

The American National Standards Institute (ANSI) has established a standard for the BASIC language but it is somewhat limited and most versions of BASIC have many additions and embellishments. We will intentionally use a primitive dialect to ensure compatibility with most modern implementations and to simplify the translation into incompatible versions. In particular the programs in this book should run without modification under Microsoft BASICA GW-BASIC QBASIC QuickBASIC VisualBASIC for MS-DOS; Borland International TurboBASIC (no longer available); and Spectra Publishing PowerBASIC on IBM PCs or compatibles. You will be happiest using a modern compiled BASIC such as VisualBASIC or PowerBASIC on a fast computer with a math coprocessor.

Appendix C includes information on translating the computer programs into other partially incompatible dialects of BASIC as well as source code for use with VisualBASIC for Windows and Microsoft QuickBASIC for the Macintosh. Appendix D contains a translation into Microsoft QuickC. The BASIC programs use line numbers which have been obsolete since the mid 1980's but they are harmless and they provide a convenient way to reference lines of the program and to indicate where in the program a change is to be made.

If you follow sequentially through this book you will only need to add and change a few lines of the program as you meet each new idea. Your program will gradually grow more versatile as you work through the book. In the end you will have a powerful program that can reproduce all the examples in this book as well as an endless variety of new ones. Hence you should avoid the temptation to eliminate or to change the line numbers at least until you have a fully functional program. You may prefer to jump to Appendix B where you will find the complete final program which is also provided on the accompanying disk along with source listings in BASIC Microsoft QuickC Borland TurboC++ and a ready-to-run executable version of the program.

If you are an experienced programmer you might ridicule some of the quaint program listings. Many powerful programming structures such as block IF statements DO LOOPs and callable subroutines with local variables that produce beautifully structured programs are now standard but they have been avoided to allow backwards compatibility with more primitive versions of BASIC. They also often impose a small speed penalty. The dreaded GOTO statement has been used primarily to bypass blocks of code in deference to BASIC versions that don't support block IF statements. Lines of the program that are bypassed by a GOTO are usually indented. Blocks of the program contained within FOR...NEXT loops have also been indented. In the interest of structure and simplicity the programs have been written using numerous small modular subroutines each with a single entry point beginning with a comment line and a single exit point containing a RETURN statement albeit with global variables. The individual subroutines are separated with blank lines. It should be relatively easy for an experienced programmer to rewrite the program in a more modern format.

The program listing PROG01 iterates the logistic equation for R = 4 with an initial value of X = 0.05 and makes a graph of each iterate versus its predecessor. The program looks more complicated than it is because the various operations have been relegated to subroutines to provide a template for the more versatile cases to follow.

PROG01 Program for Iterating and Graphing the Logistic Equation

1000 REM LOGISTIC EQUATION

1010 DEFDBL A-Z                 'Use double precision
1030 SM% = 12                   'Assume VGA graphics
1190 GOSUB 1300                 'Initialize
1200 GOSUB 1500                 'Set parameters
1210 GOSUB 1700                 'Iterate equations
1220 GOSUB 2100                 'Display results
1230 GOSUB 2400                 'Test results
1240 ON T% GOTO 1190
1200
1210
1250 CLS
1260 END
1300 REM Initialize
1320 SCREEN SM%                 'Set graphics mode
1350 WINDOW (-.1
-.1)-(1.1
1.1)
1360 CLS
1420 RETURN
1500 REM Set parameters
1510 X = .05                    'Initial condition
1560 R = 4                      'Growth rate
1570 T% = 3
1590 LINE (-.1
-.1)-(1.1
1.1)

B
1630 RETURN
1700 REM Iterate equations
1720 XNEW = R * X * (1 - X)
2030 RETURN
2100 REM Display results
2300 PSET (X
XNEW)             'Plot point on screen
2320 RETURN
2400 REM Test results
2490 IF LEN(INKEY$) THEN T% = 0 'Respond to user key stroke
2510 X = XNEW                   'Update value of X
2550 RETURN

If when you first run the program your computer reports an error it will probably be in one of the following lines:

Line 1010: Be sure your version of BASIC supports double-precision (four-byte) floating-point variables. If it doesn't you may omit this line but you will then probably have to change the 4 in line 1560 to 3.99999 to avoid overflow resulting from round-off errors. With modern versions of BASIC and a computer with a math coprocessor there is no penalty and considerable advantage in using double precision. Because of the finite precision of computer arithmetic all cases will eventually repeat but with double precision the average number of iterations required before this happens is acceptably large.

Line 1320: Either your version of BASIC doesn't require this command or your computer or compiler doesn't support VGA graphics. Try reducing the 12 in line 1030 to a lower number until you find one that works. If none work try eliminating line 1320 altogether.

Line 1350: The WINDOW command defines the coordinates of the lower-left and upper-right corners of the graphics window for subsequent PSET and LINE commands. If your version of BASIC doesn't support this command you will have to delete this line and convert all the parameters in the PSET and LINE commands to address screen pixels. In such a case try replacing line 2300 with PSET (200 * X 200 - 200 * XNEW). One advantage of using the WINDOW command is that when a version of BASIC comes along that supports higher screen resolutions the program can be easily recompiled to take advantage of it.

Other errors: Look carefully for typographical errors or consult your BASIC manual to determine compatibility.

The correct program should produce a plot of the logistic parabola as show in Figure 1-3. Try different initial values of X (line 1510) and different values of R (line 1560) to confirm the behavior predicted for the logistic equation.

AMu%

Figure 1-3. The Logistic Parabola from PROG01

The logistic parabola comes from a chaotic solution but it doesn't look very complicated and it would hardly qualify as art. With one small change we can make things more interesting while at the same time illustrating the sensitivity to initial conditions. Instead of plotting each iterate versus its immediate predecessor we could plot it versus its second or third or fourth predecessor. Let's save the last 500 iterates and provide the option to plot X versus any one of them.

The changes that you need to make in the program PROG01 to accomplish this are shown in the listing PROG02. You can either go through the program and change or add lines as necessary or type the listing and save it in ASCII format and then use the MERGE command that is supported by many (mostly old) versions of BASIC to update the previous version of the program.

PROG02 Changes Required in PROG01 to Plot the Fifth Previous Iterate

1000 REM LOGISTIC EQUATION (5th Previous Iterate)

1020 DIM XS(499)
1040 PREV% = 5                  'Plot versus fifth previous iterate
1580 P% = 0
2210 XS(P%) = X
2220 P% = (P% + 1) MOD 500
2230 I% = (P% + 500 - PREV%) MOD 500
2300 PSET (XS(I%)
XNEW)                'Plot point on screen

With PREV% = 1 in line 1040 the result should be the same as for PROG01. However if you set PREV% equal to 2 you will see the logistic parabola change into a curve with two humps. Each time you increase PREV% by one you double the number of humps in the curve. Thus with PREV% = 5 the result as shown in Figure 1-4 has sixteen oscillations.

AMu%

Figure 1-4. The Logistic Parabola after five iterations from PROG02

Figure 1-4 provides a good graphical illustration of the sensitivity to initial conditions. The horizontal axis represents all possible initial conditions from zero to one. The vertical axis shows the value from zero to one corresponding to each initial condition after five iterations. It's not hard to see that two nearby points on the horizontal axis usually translate into two very different values along the vertical axis after five iterations. Try using PREV% = 10 and convince yourself that information about the initial condition is almost completely lost after ten iterations.

This exercise provides a good insight into the way a strange attractor is formed geometrically. The logistic parabola which began as a line (a one-dimensional object) is stretched and folded with each iteration eventually filling the entire plane (a two-dimensional object) after many iterations. Perhaps it reminds you of those taffy machines that repeatedly stretch and fold the taffy causing two nearby specks in the taffy after a while to be nowhere near one another. On average the distance between the specks initially increases at an exponential rate.

You should be able to think of many other examples of the sensitivity to initial conditions. When you stir your coffee to mix in the cream you're relying on a chaotic process. Two sticks dropped into the water close together just above a waterfall eventually end up far apart. Try laying two identical garden hoses side by side and turn on the water in each one at the same time without holding the ends. Chaotic processes are all around us. Their mathematical solutions usually produce chaotic strange attractors whose diversity and beauty we are about to explore.

CHAPTER 2

Wiggly Lines

In this chapter we will teach the computer to search for chaotic solutions of simple equations with a single variable. The solutions are segments of lines but the lines can wiggle in an incredibly complicated manner.

2.1 More Knobs to Twiddle

The logistic equation (Equation 1C) is an example of a dynamical system. Such systems are described by deterministic initial-value equations. This particular system has a single parameter R whose value determines the solution's behavior for all initial values of X within the basin of attraction. It's like a knob on a radio or on a stove that you can turn up or down to control the sound emitted by the radio or the convection in a pot of boiling soup.

There is a simple experiment you can do to observe the period-doubling route to chaos. Go into your bathroom or kitchen and turn on the tap but only slightly to produce a regular periodic pattern of drips. Now slowly open the tap until the pattern becomes chaotic. Just before the onset of chaos if you are sufficiently careful and patient you should observe one or more period doublings where the sound changes to something like--drip drip - drip drip - drip drip. The knob that controls the flow rate corresponds to the parameter R in the logistic equation. The dripping faucet has been extensively studied by Robert Shaw and discussed at length in his book The Dripping Faucet as a Model Chaotic System .

Most dynamical systems have more than one knob. Your kitchen faucet probably has independent control of the flow rate and the temperature of the water. With more knobs you might expect to increase the variety of ways the system can behave. Such knobs are called control parameters.

The formula for the most general one-dimensional quadratic iterated map is

Xn+1 = a1 + a2Xn + a3Xn2 (Eq. 2A)

Here we have three control parameters--a1 a2 and a3. By exploring all combinations of their values we expect eventually to observe every possible peculiar solution that the equation can have.

You might think that the initial condition X0 is a fourth knob but if the system is chaotic the solution is generally a strange attractor and all initial conditions within the basin of attraction look the same after many iterations. Of course there is no guarantee that a particular choice of X0 lies within the basin but values of X0 close to zero are within the basin about half the time and there are so many chaotic solutions over the range of the other three parameters that we can well afford to discard half of them.

The search for strange attractors proceeds as follows. Choose values for a1 a2 and a3 arbitrarily. Start with a value of X0 near zero. Iterate Equation 2A repeatedly until the solution either exceeds some large number in which case it is presumably unbounded or until the Lyapunov exponent becomes small or negative in which case the solution is probably a fixed point or limit cycle. In either event choose a different combination of a1 a2 and a3 and start over. If after a few thousand iterations the solution is bounded (X is not enormous) and the Lyapunov exponent is positive then it is likely that you have found a strange attractor.

2.2 Randomness and Pseudo-randomness

To choose values of a1 a2 and a3 we can use the random number generator provided with most computer languages. The random numbers thus produced are usually uniformly distributed between zero and one. You may wonder how a computer the epitome of determinism could ever produce a random number. This question deserves a digression since the answer provides yet another example of the very issues we have been discussing.

One way to produce a random number is to start with a value of X (the seed) between zero and one and iterate the logistic equation with R = 4 a hundred or so times. The result is a new number in the range of zero to one that is related to the seed in a complicated and sensitive way. This number is then used as the seed for the next random number which is produced in the same way. A given seed will produce the same sequence of random numbers but the sequence may not be the same on different computers or with different languages or even with different versions of the same language because of the way the numbers are rounded.

However this method of producing random numbers is not optimal. First the numbers are not uniformly distributed over the range. They tend to cluster near zero and one as the darkness of the right-hand side of Figure 1-1 suggests. Also multiplying a non-integer number by itself a hundred times is a relatively slow process on a computer.

Instead computers usually get their random numbers using the linear congruential method:

Xn+1 = (aXn + b) mod c (Eq. 2B)

where the mod (modulus) operation means to divide the quantity on its left (aXn + b) by the quantity on its right (c) and keep the remainder rather than the quotient. All the quantities in Equation 2B are integers. The constants a b and c are carefully chosen to maximize the number of steps required for the sequence to repeat which in any case can never exceed c. The numbers are uniformly distributed from zero to c - 1 but they can be transformed to the range zero to one by simply dividing Xn+1 by c. The numbers appear to be random but since they are produced using a deterministic procedure they are often called pseudo-random. Equation 2B is another example of a one-dimensional chaotic map called a shift map.

There are infinitely many conditions that truly random numbers should satisfy. Not only must the numbers be uniform over the interval but there should be no detectable relation between the numbers and any of their predecessors. In particular the sequence should repeat only after a very large number of steps. Most random number generators are deficient in certain ways. For example the random numbers produced by Microsoft QuickBASIC 4.5 repeat after 16 777 216 steps and this number is too small for some of our purposes.

The situation can be greatly improved by shuffling the numbers. Suppose we maintain a table of a hundred or so random numbers. When we want one we randomly take an entry from the table and replace it with a new random number. With this simple modification the pseudo-random numbers generated by the computer are sufficiently random for all our needs.

You should always remember that the sequence of random numbers generated by a digital computer will eventually repeat. You must take care to ensure that over the duration of a calculation such a repetition does not occur. You must also reseed the random number generator using a truly random seed such as one based on the time of day the program is started if you are to avoid repeating the same sequence each time you run the program.

2.3 What's in a Name?

When we begin to choose random values for the coefficients a1 a2 and a3 we are immediately confronted with two issues. The first is the range of values that the coefficients may have and the second is the amount by which two values of a coefficient must differ to produce attractors that are visibly different.

The first issue can be addressed by referring to the logistic equation (Equation 1C). When the value of R is too small (less than about 3.5) there are no chaotic solutions and when the value of R is too large (greater than 4) all the solutions are unbounded. A similar situation occurs for the more general one-dimensional quadratic map in Equation 2A. Thus we want to limit the coefficients to values whose magnitude (positive or negative) is of order unity. That is to say 0.1 is probably too small a value and 10 is probably unnecessarily large. This assumption can be verified by numerical experiment.

The second issue requires a subjective judgment of how dissimilar two attractors must look before we consider them to be different. In practice a change in one of the coefficients by an amount of order 0.1 will generally produce an object that is noticeably different. If we let each coefficient take on values ranging from -1.2 to 1.2 in steps of 0.1 we will have 25 possible values and we can associate each with a letter of the alphabet A through Y and have a convenient way to catalog and replicate the attractors. Limiting the coefficients to 25 values may seem excessively restrictive but since there are three coefficients for one-dimensional quadratic maps there are 253 or 15 625 different combinations.

The coefficients that correspond to the logistic equation with R = 4 are a1 = 0 a2 = 4 and a3 = -4 and they fall outside the range of -1.2 to 1.2. Thus for some purposes it will be convenient to take a larger range. A convenient way to extend the range is to use the ASCII (American Standard Code for Information Interchange) character set summarized in Table 2.1.

Table 2.1 ASCII Character Set and Associated Coefficient Values

Code Dec Coeff Code Dec Coeff Code Dec Coeff

32 -4.5 # 64 -1.3 ` 96 1.9

! 33 -4.4 A 65 -1.2 a 97 2.0

" 34 -4.3 B 66 -1.1 b 98 2.1

# 35 -4.2 C 67 -1.0 c 99 2.2

$ 36 -4.1 D 68 -0.9 d 100 2.3

% 37 -4.0 E 69 -0.8 e 101 2.4

& 38 -3.9 F 70 -0.7 f 102 2.5

' 39 -3.8 G 71 -0.6 g 103 2.6

( 40 -3.7 H 72 -0.5 h 104 2.7

) 41 -3.6 I 73 -0.4 i 105 2.8

* 42 -3.5 J 74 -0.3 j 106 2.9

+ 43 -3.4 K 75 -0.2 k 107 3.0

44 -3.3 L 76 -0.1 l 108 3.1

- 45 -3.2 M 77 0.0 m 109 3.2

. 46 -3.1 N 78 0.1 n 110 3.3

/ 47 -3.0 O 79 0.2 o 111 3.4

0 48 -2.9 P 80 0.3 p 112 3.5

1 49 -2.8 Q 81 0.4 q 113 3.6

2 50 -2.7 R 82 0.5 r 114 3.7

3 51 -2.6 S 83 0.6 s 115 3.8

4 52 -2.5 T 84 0.7 t 116 3.9

5 53 -2.4 U 85 0.8 u 117 4.0

6 54 -2.3 V 86 0.9 v 118 4.1

7 55 -2.2 W 87 1.0 w 119 4.2

8 56 -2.1 X 88 1.1 x 120 4.3

9 57 -2.0 Y 89 1.2 y 121 4.4

: 58 -1.9 Z 90 1.3 z 122 4.5

; 59 -1.8 [ 91 1.4 { 123 4.6

< 60 -1.7 \ 92 1.5 | 124 4.7

= 61 -1.6 ] 93 1.6 } 125 4.8

> 62 -1.5 ^ 94 1.7 ~ 126 4.9

? 63 -1.4 _ 95 1.8  127 5.0

ASCII codes from 0 to 31 are reserved for control codes--things like backspace carriage return and line feed. Codes from 128 to 255 can also be used but there is no universal character set associated with them. By making use of all the ASCII characters from 0 to 255 we can accommodate coefficients in the range of -7.7 to 17.8. The characters listed in the table will suffice for most of our needs however.

With such a coding scheme each attractor is represented by a sequence of characters with each character corresponding to one of the coefficients. The sequence can be thought of as the name of the attractor. We will preface the name with a character that indicates the type of equation we are dealing with. Let's use the letter A to represent one-dimensional quadratic maps. Thus the logistic equation coded in this way is AMu%. Note that the letters in the name are case sensitive (u and U are different) and so you should be careful when typing them. Such names may look strange which is perhaps appropriate for strange attractors and you shouldn't try to pronounce them! However they do provide a convenient and compact method for saving everything you need to reproduce an attractor.

2.4 The Computer Search

Before embarking on a search for strange attractors we need to generalize the formula for the Lyapunov exponent given in Equation 1E for the logistic equation. The generalization is easily obtained using differential calculus and the result is

L = S log2 |a2 + 2a3Xn| / N (Eq. 2C)

The program changes that are required to perform a search for strange attractors in one-dimensional quadratic iterated maps are given in the listing PROG03.

PROG03 Changes Required in PROG02 to Search for Strange Attractors in One-Dimensional Quadratic Maps

1000 REM ONE-D MAP SEARCH

1020 DIM XS(499)
A(504)
V(99)
1050 NMAX = 11000               'Maximum number of iterations
1160 RANDOMIZE TIMER            'Reseed random number generator
1360 CLS : LOCATE 13
34: PRINT "Searching..."
1560 GOSUB 2600                 'Get coefficients
1580 P% = 0: LSUM = 0: N = 0: NL = 0
1590 XMIN = 1000000!: XMAX = -XMIN
1720 XNEW = A(1) + (A(2) + A(3) * X) * X
2020 N = N + 1
2110 IF N < 100 OR N > 1000 THEN GOTO 2200
2120    IF X < XMIN THEN XMIN = X
2130    IF X > XMAX THEN XMAX = X
2140    YMIN = XMIN: YMAX = XMAX
2200 IF N = 1000 THEN GOSUB 3100        'Resize the screen
2250 IF N < 1000 OR XS(I%) <= XL OR XS(I%) >= XH OR XNEW <= XL OR XNEW >= XH THEN GOTO 2320
2410 IF ABS(XNEW) > 1000000! THEN T% = 2    'Unbounded
2430 GOSUB 2900                 'Calculate Lyapunov exponent
2460 IF N >= NMAX THEN T% = 2   'Strange attractor found
2470 IF ABS(XNEW - X) < .000001 THEN T% = 2 'Fixed point
2480 IF N > 100 AND L < .005 THEN T% = 2    'Limit cycle
2600 REM Get coefficients
2660 CODE$ = "A"
2680 M% = 3
2690    FOR I% = 1 TO M%        'Construct CODE$
2700        GOSUB 2800          'Shuffle random numbers
2710        CODE$ = CODE$ + CHR$(65 + INT(25 * RAN))
2720    NEXT I%
2730 FOR I% = 1 TO M%           'Convert CODE$ to coefficient values
2740    A(I%) = (ASC(MID$(CODE$
I% + 1
1)) - 77) / 10
2750 NEXT I%
2760 RETURN
2800 REM Shuffle random numbers
2810 IF V(0) = 0 THEN FOR J% = 0 TO 99: V(J%) = RND: NEXT J%
2820 J% = INT(100 * RAN)
2830 RAN = V(J%)
2840 V(J%) = RND
2850 RETURN
2900 REM Calculate Lyapunov exponent
2910 DF = ABS(A(2) + 2 * A(3) * X)
3030 IF DF > 0 THEN LSUM = LSUM + LOG(DF): NL = NL + 1
3040 L = .721347 * LSUM / NL
3070 RETURN
3100 REM Resize the screen
3120 IF XMAX - XMIN < .000001 THEN XMIN = XMIN - .0000005: XMAX = XMAX + .0000005
3130 IF YMAX - YMIN < .000001 THEN YMIN = YMIN - .0000005: YMAX = YMAX + .0000005
3160 MX = .1 * (XMAX - XMIN): MY = .1 * (YMAX - YMIN)
3170 XL = XMIN - MX: XH = XMAX + MX: YL = YMIN - MY: YH = YMAX + MY
3180 WINDOW (XL
YL)-(XH
YH): CLS
3310 LINE (XL
YL)-(XH
YH)

B
3460 RETURN

There are several things to note about the listing PROG03:

1. The maximum number of iterations (NMAX in line 1050) has been set arbitrarily to 11 000. This is the number of iterations after which a strange attractor is assumed to have been found if the magnitude of X is less than one million and the Lyapunov exponent is positive (actually greater than 0.005). You may wish to decrease NMAX to speed the rate at which attractors are found or increase NMAX if you have a very fast computer or want to give the displays more time to develop. The number of iterations is a parameter that you can adjust for the most visually appealing result. Most of the figures in this book were made with NMAX between about half a million and ten million and required between about a minute and an hour to produce.

2. The seed for the random number generator is taken in line 1160 as the number of seconds lapsed since midnight (TIMER). This choice ensures that a new sequence of random numbers is produced each time the program is run except in the unlikely event that it is run at exactly the same time each day.

3. After 1000 iterations (line 2200) the screen is resized and erased by the subroutine in lines 3100 through 3460 using the minimum and maximum values of X between the hundredth and thousandth iteration allowing a ten percent border around the attractor.

4. To save time the difference between each value of X and its predecessor is evaluated in line 2470 and if the difference is less than one millionth the solution is assumed to be a fixed point even if the Lyapunov exponent is still positive.

5. The Lyapunov exponent is not used as a criterion until after 100 iterations (line 2480) to ensure that its value is reasonably accurate.

6. The coefficients of the equation are chosen in line 2710 using random numbers which have been shuffled by the subroutine in lines 2800 though 2850 to minimize the chance of repeating the same search sequence.

The criterion for detecting a strange attractor is somewhat subjective. There will always be borderline cases for which no amount of computing will suffice to distinguish between a strange attractor and a periodic solution with a very long period. However our interest here is in finding visually interesting attractors quickly and so we can afford to make occasional mistakes. Such mistakes account for only a small number of cases.

Of the 15 625 combinations of coefficients exactly 364 (2.3%) of them are chaotic by these criteria. Some of the more visually interesting ones are shown in Figures 2-1 through 2-4 in which the values are plotted versus their fifth previous iterate. For each case the code and the Lyapunov exponent are shown at the top of the graph.

AXBH

Figure 2-1. One-Dimensional Quadratic Map

ABDU

Figure 2-2. One-Dimensional Quadratic Map

ACAV

Figure 2-3. One-Dimensional Quadratic Map

AXDA

Figure 2-4. One-Dimensional Quadratic Map

The search for strange attractors is potentially time-consuming if you have an old computer without a math coprocessor or if you are using a BASIC interpreter rather than a compiler. Even if the search is reasonably fast on your computer be forewarned that it will slow down considerably as you advance to the more complicated equations later in the book. Perhaps this is a good time to summarize some of your options for making the program run faster.

When comparing calculation speeds of various computers and compilers it is important to do the comparison with the actual program or with a benchmark that accurately reflects its mix of instructions graphics and disk access. With computer speeds doubling approximately every two years speed will eventually cease to be a consideration for the calculations described in this book. Meanwhile you need to consider the alternatives.

Table 2.2 lists the average number of strange attractors found by PROG03 per hour using various versions of BASIC on a 33 MHz 80486DX-based computer with and without a math coprocessor. The exact numbers are less important than the relative values. They provide a good indication of how the various versions of BASIC compare on calculations of the type that will be used throughout this book.

Table 2.2 Strange Attractors Found per Hour by PROG03 with Various Versions of BASIC

QuickBASIC and VisualBASIC for MS-DOS can be run from the editor environment where they function much like an interpreter or they can be used to compile a stand-alone executable program. VisualBASIC can be compiled with either of two floating point math packages; the alternate package is faster for machines without a coprocessor and the emulate package is faster for machines with a coprocessor. TurboBASIC is now obsolete and has been replaced by PowerBASIC. PowerBASIC like VisualBASIC can be compiled with either of two floating point math packages; the procedure package is similar to the VisualBASIC alternate package. A third math package NPX (87) is the same as emulate except that it will not work on a machine without a math coprocessor. The tests were done with all error trapping turned off which is inadvisable until you have a thoroughly debugged program.

If you launch the program from Microsoft Windows you might find the computation speeds considerably different from those in Table 2.2. In one test the PowerBASIC speeds were cut in half and the QuickBASIC speeds were increased slightly from the values obtained when the program was run directly from DOS. You should do your own speed tests to see what configuration provides the optimum performance on your computer and operating system.

The executable program on the disk that accompanies this book was compiled with PowerBASIC using the procedure package. If you have PowerBASIC and a math coprocessor you can recompile the program using the emulate or NPX (87) package to achieve a slight improvement in speed.

2.5 Wiggles on Wiggles

The preceding figures consist of segments of wiggly lines and thus they are not very artistic. To make things more interesting we can consider one-dimensional maps of higher order. By this we mean that we will not stop with quadratic (X2) maps but we will consider equations containing cubic (X3) quartic (X4) quintic (X5) and even higher terms.

In one sense considering higher-order terms is equivalent to plotting each iterate versus an iterate earlier than the immediately previous one. For example two successive iterations of the second-order Equation 2A yields

Xn+2 = a1(1+a2+a1a3) + (a3a2+2a1a3)Xn

(Eq. 2D)

+ a3(a2+2a1a3+a22)Xn2 + 2a2a32Xn3 + a33Xn4

which is a fourth-order polynomial. However there are only three parameters a1 a2 and a3 from which the five coefficients are uniquely determined.

A simpler and more general procedure is to allow each term in the polynomial to have its own coefficient which for fifth order gives

Xn+1 = a1 + a2Xn + a3Xn2 + a4Xn3 + a5Xn4 + a6Xn5 (Eq. 2E)

With six coefficients each with 25 possible values there are 256 or about 244 million different combinations. Even if only a small percentage of them is chaotic we would have to look at one every second for about a year before we would see them all.

The generalization of the expression for the Lyapunov exponent for a fifth-order map is given by

L = S log2 |a2 + 2a3Xn + 3a4Xn2 + 4a5Xn3 + 5a6Xn4| / N (Eq. 2C)

With these equations in hand you can easily modify the program as in PROG04 to search for one-dimensional attractors of up to fifth order. In our coding scheme a first letter of B will represent third order C will represent fourth order and D will represent fifth order. The program is written so that even higher orders can be produced by changing the quantity OMAX% in line 1060.

PROG04 Changes Required in PROG03 to Search for Strange Attractors in One-Dimensional Maps of order up to OMAX%

1000 REM ONE-D MAP SEARCH (Polynomials up to 5th Order)

1060 OMAX% = 5                  'Maximum order of polynomial
1720 XNEW = A(O% + 1)
1730 FOR I% = O% TO 1 STEP -1
1830    XNEW = A(I%) + XNEW * X
1930 NEXT I%
2650 O% = 2 + INT((OMAX% - 1) * RND)
2660 CODE$ = CHR$(63 + O%)
2680 M% = O% + 1
2910 DF = 0
2930 FOR I% = O% TO 1 STEP -1
2940    DF = I% * A(I% + 1) + DF * X
2970 NEXT I%
3000 DF = ABS(DF)

The listing in PROG04 produces a more interesting array of shapes samples of which are shown in Figures 2-5 through 2-10. The objects are still segments of lines but the wiggles themselves have wiggles and the underlying determinism is less obvious than before.

BZEZK

Figure 2-5. One-Dimensional Cubic Map

CBLCTX

Figure 2-6. One-Dimensional Quartic Map

CUTXJE

Figure 2-7. One-Dimensional Quartic Map

DBOGIZI

Figure 2-8. One-Dimensional Quintic Map

DFBIEVV

Figure 2-9. One-Dimensional Quintic Map

DOOYRIL

Figure 2-10. One-Dimensional Quintic Map

2.6 Making Music

If the preceding figures don't qualify as art perhaps they qualify as music. Since the quantity X behaves in a deterministic yet unpredictable way it may be that a sequence of musical notes determined by X will mimic the order and unpredictability that characterize music. It's easy to test.

Suppose we allow the notes to span three octaves from A-220 to A-1760. The letter refers to the musical note and the numbers refer to the frequency in cycles per second (called Hertz). We'll allow the notes to take one of twelve distinct values corresponding to the even-tempered scale and for simplicity take all the notes to be of the same duration. Thus the range of possible values of X is divided into 36 intervals and each successive iterate of X is converted into the corresponding musical note. The listing PROG05 shows the changes necessary to accomplish this.

PROG05 Changes Required in PROG04 to Produce Chaotic Music

1000 REM ONE-D MAP SEARCH (With Sound)

1100 SND% = 1                   'Turn sound on
2310    IF SND% = 1 THEN GOSUB 3500     'Produce sound
2490 Q$ = INKEY$: IF LEN(Q$) THEN GOSUB 3600        'Respond to user command
3500 REM Produce sound
3510 FREQ% = 220 * 2 ^ (CINT(36 * (XNEW - XL) / (XH - XL)) / 12)
3520 DUR = 1
3540 SOUND FREQ%
DUR: IF PLAY(0) THEN PLAY "MF"
3550 RETURN
3600 REM Respond to user command
3610 T% = 0
3630 IF ASC(Q$) > 96 THEN Q$ = CHR$(ASC(Q$) - 32)
3770 IF Q$ = "S" THEN SND% = (SND% + 1) MOD 2: T% = 3
3800 RETURN

The program allows you to toggle the sound on and off by pressing the S key. Pressing any other key exits the program. You might wish to experiment with the duration DUR of the SOUND statement in line 3520. Increasing its value from 1 (corresponding to approximately 0.055 seconds) makes the sounds more musical but the calculation will then take longer.

The use of sound to help interpret data generated by a computer is a technique that is relatively unexplored. The method is sometimes called sonification. In some cases patterns and structure in data can be more readily discerned audibly than visually. This technique was used to advantage in interpreting data from the Voyager spacecraft as it detected plasma waves near Jupiter and micrometeorites as it crossed through the rings of Saturn. The repetitive sound of a simple limit cycle contrasts sharply with the non-repetitive waverings of a chaotic time series.

CHAPTER 3

Pieces of Planes

Whereas the last chapter dealt with one-dimensional maps whose graphs are segments of lines this chapter will deal with two-dimensional maps whose graphs are pieces of planes and which thus produce much more interesting displays. This chapter provides the minimum tools for creating attractors that genuinely qualify as art. With only the information contained here the variety of available patterns is so great that you hardly need to proceed beyond this chapter but if you do stop here you will miss some delightful surprises.

3.1 Quadratic Maps in Two Dimensions

In the discussion so far the maps have involved a single variable X whose value changes with each iteration of the equation. Such maps are said to be one-dimensional because the values of X can be thought of as lying along a line and a line is a one-dimensional object. By plotting each value of X versus a previous value of X the line can be made to wiggle with considerable complexity but it always remains a line and lines are of limited interest and beauty.

The situation is more interesting if we consider iterated maps that involve two variables X and Y. In such a case each iterate produces a point in a plane where X by convention represents the horizontal coordinate and Y represents the vertical coordinate of the point. With successive iteration the points fill in some portion of the plane. The visually interesting cases as usual are the chaotic ones.

Such two-dimensional maps might arise for example from an ecological model only slightly more complicated than the logistic equation. A classic example is the predator-prey problem in which X represents the prey and Y the predator. In a simple linear model the solution is a fixed point (a unique number of rabbits and foxes) or a limit cycle (the number of rabbits and foxes oscillate reaching their maximum values at different times but eventually repeating). When nonlinear terms are introduced into the model the population of each species can behave chaotically. You can think of each point that makes up such an attractor as the population of rabbits and foxes in successive years. Since such complexity arises from these very simple models it's easy to understand why ecologists might have trouble predicting the fate of biological species!

Perhaps the best known chaotic two-dimensional map is the Hénon map (proposed by the French astronomer Michel Hénon in 1976) whose equations are

Xn+1 = 1 + aXn2 + bYn

(Eq. 3A)

Yn+1 = Xn

The quantities a and b are the control parameters analogous to R in the logistic equation. Hénon used the values a = -1.4 and b = 0.3. The necessary nonlinearity is provided by the X2-term in the first equation. The Hénon map is special in that the net contraction of a set of initial points covering an area of the XY-plane is constant with each iteration. The area occupied by the points is 30% of the area at the previous iteration (from the bYn-term). Other values of b can be used but not all values produce chaotic solutions. Unlike the logistic map the Hénon map is invertable; there is a unique value for Xn and Yn corresponding to each Xn+1 and Yn+1. You may have seen an alternate form of the Hénon equations in which the factor b appears instead in the second equation. The result of repeated iteration of Equations 3A is shown in Figure 3-1.

EWM?MPMM

Figure 3-1. The Hénon Map

The resulting graph is more than a line but less than a surface. What resembles a single line is a pair of lines each one of which in turn is another pair of lines and so forth to however close you look or whatever magnification you choose. This self-similarity is a common characteristic of a class of objects that are called fractals.

Fractals are to chaos what geometry is to algebra--the visual expression of the mathematical idea. Approaching an understanding of chaos through such visual means is appealing to those with an aversion to conventional mathematics. The Euclidean geometry that we learned in high school originated with the ancient Greeks and was developed more fully by the French mathematician Descartes and others in the 1600's. It deals with simple shapes such as lines circles and spheres. Euclidean geometry is now being augmented by fractal geometry whose father and champion is the contemporary mathematician Benoit Mandelbrot. Fractals appeared in art such as in the drawings of the Dutch artist Maurits C. Escher before they were widely appreciated by mathematicians and scientists.

Some fractals are exactly self-similar which means that they look the same no matter how much you magnify them. Others such as most of the ones in this book only have regions that are self-similar. There is no part of the Hénon attractor where you can zoom in and find a miniature replica of the entire attractor. Other fractals are only statistically self-similar which means that a magnified portion of the object has the same amount of detail as the whole but it is not an exact replica of it. Nearly all strange attractors are fractals but not all fractals arise from strange attractors.

The Hénon map produces an object with a fractal dimension that is a fraction intermediate between one and two. The fractal dimension is a useful quantity for characterizing strange attractors. Isolated points have dimension zero line segments have dimension one surfaces have dimension two and solids have dimension three. Strange attractors generally have non-integer dimensions.

Since the Hénon map has X2 as its highest-order term it is a quadratic map. The most general two-dimensional quadratic iterated map is

Xn+1 = a1 + a2Xn + a3Xn2 + a4XnYn + a5Yn + a6Yn2

(Eq. 3B)

Yn+1 = a7 + a8Xn + a9Xn2 + a10XnYn + a11Yn + a12Yn2

Equations 3B have twelve coefficients. For the Hénon map a1 = 1 a3 = -1.4 a5 = 0.3 a8 = 1 and the other coefficients are zero. If we use the initial letter E to represent two-dimensional quadratic maps the code for the Hénon map according to Table 2.1 is EWM?MPM2WM4 where we have introduced the shorthand M2 for MM and M4 for MMMM.

Values of a in the range of -1.2 to 1.2 are sufficient to produce an enormous variety of strange attractors. With increments of 0.1 there are 2512 or about 6 x 1016 different cases of which approximately 1.6% or about 1015 are chaotic. Viewing them all at a rate of one per second would require over 30 million years! Stated differently if each one were printed on an 8 1/2 by 11-inch sheet of paper the collection would cover nearly the entire land mass of the Earth.

Note that not all the cases are strictly distinct. For example if you replace X with Y and Y with -X in Equations 3B you will produce an attractor rotated 90 degrees counter-clockwise from the original. When you do this be sure to change Xn+1 and Yn+1 as well as Xn and Yn. Thus the code EM4CMWJM3? will produce a rotated version of the Hénon map. In the same fashion you can rotate an attractor through 180 degrees by replacing X with -X and Y with -Y and through 270 degrees by replacing X with -Y and Y with X . Perhaps it's easier just to rotate your computer monitor!

Besides rotations there are cases that correspond to reflections. When viewed in a mirror the attractors have left and right reversed but up and down remain the same. A transformation in which X is replaced with -X will accomplish this. Thus the code for a reflected Hénon map is ECM[MJM2CM4. In addition the reflections can be rotated. Thus there are at least eight so-called degenerate states for each attractor corresponding to rotations and reflections. Such symmetries and degeneracies play an important role in science; they often reduce the amount of work we have to do and provide relations between phenomena that initially appear different.

There are additional degenerate cases corresponding to scale changes. For example if you replace X by mX and Y by nY with m = n the attractor remains the same except it is reduced in size by a factor of m. Some of the coefficients are likely to be outside the allowed range however. The Hénon map with m = n = 2 can be generated with the code ERM1MPM2WM4. With m not equal to n the horizontal and vertical dimensions are scaled differently but since the computer rescales the attractor to fit the screen the visual result is the same.

These degeneracies show that there are many ways to code a particular attractor. Although this is true there are so many different possible combinations of coefficients that it is very unlikely that two degenerate cases will be found spontaneously. Thus the examples displayed in this chapter represent but a tiny fraction of the possibilities and you will be generating many other cases almost none of which have been seen before.

3.2 The Butterfly Effect Revisited

Two-dimensional chaotic iterated maps also exhibit sensitivity to initial conditions but the situation is more complicated than for one-dimensional maps. Imagine a collection of initial conditions filling a small circular region of the XY-plane. After one iteration the points will have moved to a new position in the plane but they will now occupy an elongated region called an ellipse. The circle will have contracted in one direction and expanded in the other. With each iteration the ellipse will get longer and narrower eventually stretching out into a long filament. The orientation of the filament also changes with each iteration and it wraps up like a ball of taffy.

Thus two-dimensional chaotic maps have not a single Lyapunov exponent but two--a positive one corresponding to the direction of expansion and a negative one corresponding to the direction of contraction. The signature of chaos is that at least one of the Lyapunov exponents is positive. Furthermore the magnitude of the negative exponent has to be greater than the positive one so that initial conditions scattered throughout the basin of attraction contract onto an attractor that occupies a negligible portion of the plane. The area of the ellipse continually decreases even as it stretches to an infinite length.

There is a proper way to calculate both of the Lyapunov exponents. For the mathematically inclined the procedure involves summing the logarithms of the eigenvalues of the Jacobian matrix of the linearized transformation with occasional Gram-Schmidt reorthonormalization. This method is slightly complicated and so we will instead devise a simpler procedure sufficient for determining the largest Lyapunov exponent which is all we need to test for chaos.

Suppose we take two arbitrary but nearby initial conditions. The first few iterations of the map may cause the points to get closer together or farther apart depending upon the initial orientation of the two points. Eventually the points will come arbitrarily close in the direction of the contraction but they will continue to separate in the direction of the expansion. Thus if we wait long enough the rate of separation will be governed only by the largest Lyapunov exponent. Fortunately this usually takes just a few iterations.

However because the separation grows exponentially for a chaotic system the points quickly become too far apart for an accurate estimate of the exponent. This problem can be remedied if after each iteration the points are moved back to their original separation along the direction of the separation. The Lyapunov exponent is then determined by the average of the distance they must be moved for each iteration to maintain a constant small separation. If the two solutions are separated by a distance dn after the n'th iteration and the separation after the next iteration is dn+1 the Lyapunov exponent is determined from

L = S log2 (dn+1 / dn) / N (Eq. 3C)

where the sum is taken over all iterations from n = 0 to n = N-1. After each iteration the value of one of the iterates is changed to make dn+1 = dn. For the cases here dn is taken equal to 10-6. This procedure will also allow us to deal with maps of three and even higher dimension in which there are more than two Lyapunov exponents.

3.3 Searching the Plane

We now have all the tools in hand to conduct a computer search for attractors in two dimensions. The procedure is the same as for one-dimensional maps except that the Lyapunov exponent calculation is done differently and the X and Y variables are plotted as a point in the plane after each iteration. Listing PROG06 shows the changes needed to accomplish such a search.

PROG06 Changes Required in PROG05 to Search for Two-Dimensional Quadratic Strange Attractors

1000 REM TWO-D MAP SEARCH

1060 OMAX% = 2                  'Maximum order of polynomial
1070 D% = 2                     'Dimension of system
1100 SND% = 0                   'Turn sound off
1520 Y = .05
1550 XE = X + .000001: YE = Y
1590 XMIN = 1000000!: XMAX = -XMIN: YMIN = XMIN: YMAX = XMAX
1720 XNEW = A(1) + X * (A(2) + A(3) * X + A(4) * Y)
1730 XNEW = XNEW + Y * (A(5) + A(6) * Y)
1830 YNEW = A(7) + X * (A(8) + A(9) * X + A(10) * Y)
1930 YNEW = YNEW + Y * (A(11) + A(12) * Y)
2140    IF Y < YMIN THEN YMIN = Y
2150    IF Y > YMAX THEN YMAX = Y
2240 IF D% = 1 THEN XP = XS(I%): YP = XNEW ELSE XP = X: YP = Y
2250 IF N < 1000 OR XP <= XL OR XP >= XH OR YP <= YL OR YP >= YH THEN GOTO 2320
2300    PSET (XP
YP)                   'Plot point on screen
2410 IF ABS(XNEW) + ABS(YNEW) > 1000000! THEN T% = 2    'Unbounded
2470 IF ABS(XNEW - X) + ABS(YNEW - Y) < .000001 THEN T% = 2
2520 Y = YNEW
2660 CODE$ = CHR$(59 + 4 * D% + O%)
2680 M% = 1: FOR I% = 1 TO D%: M% = M% * (O% + I%): NEXT I%
2910 XSAVE = XNEW: YSAVE = YNEW: X = XE: Y = YE: N = N - 1
2930 GOSUB 1700                 'Reiterate equations
2940 DLX = XNEW - XSAVE: DLY = YNEW - YSAVE
2960 DL2 = DLX * DLX + DLY * DLY
2970 IF CSNG(DL2) <= 0 THEN GOTO 3070       'Don't divide by zero
2980    DF = 1000000000000# * DL2
2990    RS = 1 / SQR(DF)
3000    XE = XSAVE + RS * (XNEW - XSAVE): YE = YSAVE + RS * (YNEW - YSAVE)
3020    XNEW = XSAVE: YNEW = YSAVE
3030    IF DF > 0 THEN LSUM = LSUM + LOG(DF): NL = NL + 1
3040    L = .721347 * LSUM / NL
3110 IF D% = 1 THEN YMIN = XMIN: YMAX = XMAX

This program produces an incredible variety of interesting patterns a small selection of which is shown in Figures 3-2 through 3-17. You will want to admire the beauty and variety of these figures and then make some of you own by running the program and if you computer has a printer using the Print Screen key to print any that you find especially appealing.

EAGHNFOD

Figure 3-2. Two-Dimensional Quadratic Map

EBCQAFMF

Figure 3-3. Two-Dimensional Quadratic Map

EDSYUECI

Figure 3-4. Two-Dimensional Quadratic Map

EELXAPXM

Figure 3-5. Two-Dimensional Quadratic Map

EEYYMKTU

Figure 3-6. Two-Dimensional Quadratic Map

EJTTSMBO

Figure 3-7. Two-Dimensional Quadratic Map

ENNMJRCT

Figure 3-8. Two-Dimensional Quadratic Map

EOUGFJKD

Figure 3-9. Two-Dimensional Quadratic Map

EQKOCSID

Figure 3-10. Two-Dimensional Quadratic Map

EQLOIARX

Figure 3-11. Two-Dimensional Quadratic Map

ETJUBWED

Figure 3-12. Two-Dimensional Quadratic Map

ETSILUND

Figure 3-13. Two-Dimensional Quadratic Map

EUEBJLCD

Figure 3-14. Two-Dimensional Quadratic Map

EVDUOTLR

Figure 3-15. Two-Dimensional Quadratic Map

EWLKWPSM

Figure 3-16. Two-Dimensional Quadratic Map

EZPMSGCN

Figure 3-17. Two-Dimensional Quadratic Map

If you are an experienced programmer you might consider writing a screen-saver program based on PROG06. Such a terminate-and-stay-resident (TSR) program is run once when the computer is turned on and leaves a portion of itself in memory constantly monitoring keyboard and mouse activity. When there is no user activity for five minutes say it would blank the screen and begin displaying a succession of unique strange attractors to prevent screen burn-in. The original screen would be restored whenever a key is pressed or the mouse is moved. PowerBASIC version 3.0 allows you to do this easily by embedding the program within POPUP statements.

3.4 The Fractal Dimension

The previous figures differ considerably in how densely they fill the plane. Some are very thin; others are thick. A good contrast is provided by Figures 3-16 and 3-17. Figure 3-16 resembles a piece of string that has been laid down in a complicated shape on the page whereas Figure 3-17 looks like a twisted piece of paper with many holes in it. Thus the object in Figure 3-16 has a fractal dimension close to 1 and the object in Figure 3-17 has a fractal dimension closer to 2.

It is possible to be more explicit and to calculate the fractal dimension exactly. Consider two simple cases one in which successive iterates lie uniformly along a straight line that goes diagonally across the page and the other in which successive iterates gradually fill the entire plane as if they were grains of pepper sprinkled on the paper from a great height. The first case has dimension 1 and the second has dimension 2. How would we calculate the dimension given the X and Y coordinates of an arbitrary collection of such points?

One method would be to draw a small circle somewhere on the plane that surrounds at least one of the points. Then draw a second circle with the same center but with twice the radius. Now count the number of points inside each circle. Let's say the smaller circle encloses N1 points and the larger circle encloses N2 points. Obviously N2 is greater than or equal to N1 since all the points inside the inner circle are also inside the outer circle.

If the points are widely separated then N2 will equal N1. If the points are part of a straight line the larger circle will on average enclose twice as many points as the smaller circle but if the points are part of a plane the larger circle will on average enclose four times as many points as the smaller circle since the area of a circle is proportional to the square of its radius. Thus for these simple cases the dimension is given by

F = log2 (N2 / N1) (Eq. 3D)

It is hardly surprising that if you do this operation for the cases shown in the figures the quantity F is neither 0 nor 1 nor 2 but rather a fraction.

With real data a number of practical considerations determine the accuracy of the result and the amount of computation required to obtain it:

1. Is a circle the best shape or would a square rectangle triangle or some other shape be better?

2. How large should the circle be?

3. Is doubling the size of the circle optimal or would some other factor be better?

4. Where should the circles be placed and how many circles are required to obtain a representative average?

5. How many points are needed to produce a reliable fractal dimension?

Let's address each of these questions in turn.

There is nothing special about circles. A rectangle triangle or any other two-dimensional figure would suffice since the area scales as the square of the linear dimension in each case. However a circle is convenient since it is easy to tell whether a given point is in its interior by comparing the radius of the circle with the distance of the point from its center.

The optimum size of the circle represents a compromise. Ideally the circles should be invisibly small since the dimension is defined in the limit of infinite resolution. However if the circles are too small they contain too few points to produce a statistically meaningful result unless an unreasonably large number of iterations is performed. We will somewhat arbitrarily take the circles to have a radius equal to about 2% of the diagonal of the smallest rectangle that contains the attractor.

Similarly doubling the radius of the circle is arbitrary. Small values degrade the statistics and large values miss too much of the fine-scale structure. We will use a value of ten with the smaller circle about 0.6% the size of the attractor and the larger circle about 6% the size of the attractor. Thus in Equation 3D we will use logarithms of base 10 (log10) instead of base 2 (log2).

Ideally the circles should be placed uniformly or randomly over the plane. However if we were to do this most of the circles would be empty and a very long calculation would be required to obtain an accurate estimate of the dimension. Instead we center the circles on the data points themselves. In this way the circles tend to enclose many points. However it represents a different type of average since it weighs more heavily the portions of the attractor where the points are most dense. Technically what we are calculating is called the correlation dimension since it involves the number of points that are correlated with each point in the data set. The correlation dimension is never greater than the fractal dimension but it tends not to be much smaller either.

The correlation dimension is only one of many ways to define the dimension of an attractor. The various methods differ in how the regions of the attractor are weighed in the average. It is probably the easiest method to implement and it gives more reliable results than the fractal dimension when the dimension of the attractor is greater than about two. The fractal dimension is also called the capacity dimension and it is closely related to the Hausdorff-Besicovitch dimension. Furthermore the correlation dimension is probably a more meaningful measure of the strangeness of the attractor since it includes information about its formation as well as its final appearance.

The number of data points required to provide an accurate estimate of the dimension is a question still being debated in the scientific literature. Therefore we will use an heuristic approach and continually update the dimension estimate with each iteration giving you an opportunity to decide when it seems to have converged to a unique value. To do this we must modify the procedure slightly. Rather than count the number of data points within a circle which would require that the calculation run to conclusion with the coordinates of all the points saved we will use the equivalent method of determining the probability that two randomly chosen points are within a certain distance of one another. To do this the distance of each new iterate from one of its randomly chosen predecessors is calculated. Now you see why we bothered to save the last 500 iterates! We will exclude the most recent 20 points since the iterates are likely to be abnormally highly correlated with their recent predecessors. Thus with each iteration we have only one additional calculation to do in which we compare the distance of the iterate to one of its randomly chosen predecessors and increment N1 and N2 as appropriate. The listing PROG07 shows the changes needed to calculate and display the fractal dimension.

PROG07 Changes Required in PROG06 to Calculate and Display the Fractal Dimension

1000 REM TWO-D MAP SEARCH (With Fractal Dimension)

1020 DIM XS(499)
YS(499)
A(504)
V(99)
1580 P% = 0: LSUM = 0: N = 0: NL = 0: N1 = 0: N2 = 0
1620 TWOD% = 2 ^ D%
2210 XS(P%) = X: YS(P%) = Y
2440 GOSUB 3900                 'Calculate fractal dimension
3030    LSUM = LSUM + LOG(DF): NL = NL + 1
3060    IF N > 1000 AND N MOD 10 = 0 THEN LOCATE 1
76: PRINT USING "##.##"; L;
3170 XL = XMIN - MX: XH = XMAX + MX: YL = YMIN - MY: YH = YMAX + 1.5 * MY
3190 YH = YH - .5 * MY
3400 LOCATE 1
1: PRINT CODE$
3420 LOCATE 1
63: PRINT "F =": LOCATE 1
73: PRINT "L ="
3900 REM Calculate fractal dimension
3910 IF N < 1000 THEN GOTO 4010         'Wait for transient to settle
3920 IF N = 1000 THEN D2MAX = (XMAX - XMIN) ^ 2 + (YMAX - YMIN) ^ 2
3930 J% = (P% + 1 + INT(480 * RND)) MOD 500
3940 DX = XNEW - XS(J%): DY = YNEW - YS(J%)
3950 D2 = DX * DX + DY * DY
3960 IF D2 < .001 * TWOD% * D2MAX THEN N2 = N2 + 1
3970 IF D2 > .00001 * TWOD% * D2MAX THEN GOTO 4010
3980    N1 = N1 + 1
3990    F = .434294 * LOG(N2 / (N1 - .5))
4000    LOCATE 1
66: PRINT USING "##.##"; F;
4010 RETURN

At this point you might wish to examine the fractal dimension of the various figures in this book as well as the dimension of those you create with PROG07. One thing you will notice is that the dimension of objects that resemble lines is often less than 1.0. One reason is that the points that make up the line are seldom uniformly distributed along its length. Remember that the correlation dimension is usually smaller than the fractal dimension. They are equal if the points are uniformly distributed over the attractor. The correlation dimension of a line consisting of a uniform distribution of points along its length would be exactly 1.0.

Also note that the dimension of most attractors varies considerably from one part of the attractor to another. Figure 3-11 is a good example of one in which parts of the attractor resemble thin lines and other parts resemble filled-in planes. It is obviously simplistic to characterize such an object by a single average dimension.

The dimension also depends on scale. It is properly defined in the limit where one zooms in very tightly on the attractor to observe its finest detail. However a calculation in this limit would take forever since an infinite number of iterations would be required to collect enough points to reveal the detail. Figure 3-13 is an example of an attractor that is nearly one-dimensional on a large scale but closer to two-dimensional on a fine scale. Our calculation provides what might be called a visual dimension since it is taken on a scale close to what the eye can visually resolve. In any case you should not ascribe undue significance to the calculated dimension.

Also note that we are using the word "dimension" to mean several different things. The maps that we are looking at are two-dimensional because they have two variables X and Y. However the attractor has a smaller dimension. We say the attractor is embedded in a two-dimensional space or that the embedding dimension is 2. A point or a line can be embedded in a plane but a ball cannot.

An attractor usually fills a negligible portion of the space in which it is embedded. That's why it's called an attractor! Points initially distributed throughout the embedding space are drawn to the attractor after a number of iterations and the remaining space is left empty. Thus the area of an attractor embedded in a two-dimensional space is zero and the volume of an attractor embedded in a three-dimensional space is zero and so forth.

It is also interesting that the fractal dimension and the Lyapunov exponents are not entirely independent. It has been conjectured but never completely proved that the fractal dimension is related to the two Lyapunov exponents by

F = 1 - L1 / L2 (Eq. 3E)

where L1 is the more positive of the two exponents and is the one we denote by L in the figures. If both Lyapunov exponents are known Equation 3E can be used to define a dimension of the attractor called the Lyapunov dimension. The Lyapunov dimension is also called the Kaplan-Yorke dimension after the scientists who made the conjecture.

This relation is reasonable since if the two exponents are equal but of opposite signs (L2 = - L1) the contraction in one direction is just offset by expansion in the other. A set of initial conditions spread out over a two-dimensional region thus maintains its area upon successive iteration. Such a mapping is said to be area-preserving or Hamiltonian after the nineteenth century Irish astronomer William Rowan Hamilton. On the other hand if the contraction is very rapid (L2 is large and negative) the initial conditions quickly collapse to a very elongated ellipse whose dimension is close to 1. Such a contraction is sometimes called filamentation.

Armed with information about the fractal dimension you can program the computer to be even more selective. For example the visually appealing attractors tend to have fractal dimensions slightly greater than 1 and thus you could reject those with smaller dimensions or those with dimensions close to 2. We will return to this intriguing possibility in Chapter 8.

3.5 Higher Order Disorder

With one-dimensional maps the attractors became more interesting when we considered terms higher than quadratic. It is straightforward to do the same with two-dimensional maps. For example the most general equations for two-dimensional cubic maps are:

Xn+1 = a1 + a2Xn + a3Xn2 + a4Xn3 + a5Xn2Yn

+ a6XnYn + a7XnYn2 + a8Yn + a9Yn2 + a10Yn3

(Eq. 3F)

Yn+1 = a11 + a12Xn + a13Xn2 + a14Xn3 + a15Xn2Yn

+ a16XnYn + a17XnYn2 + a18Yn + a19Yn2 + a20Yn3

Note that there are twenty coefficients which vastly increases the number of possible cases. The fourth-order case would have 30 coefficients and the fifth-order case would have 42 coefficients. If you prefer an equation a two-dimensional map of order O will have (O + 1)(O + 2) coefficients. We will code the cubic quartic and quintic cases with the letters F G and H respectively.

The changes that must be made to the program to generate attractors in two dimensions up to fifth order are given in the listing PROG08.

PROG08 Changes Required in PROG07 to Generate Attractors in Two Dimensions up to Fifth Order

1000 REM TWO-D MAP SEARCH (Polynomials up to 5th Order)

1020 DIM XS(499)
YS(499)
A(504)
V(99)
XY(4)
XN(4)
1060 OMAX% = 5                  'Maximum order of polynomial
1720 M% = 1: XY(1) = X: XY(2) = Y
1730 FOR I% = 1 TO D%
1740 XN(I%) = A(M%)
1750 M% = M% + 1
1760 FOR I1% = 1 TO D%
1770 XN(I%) = XN(I%) + A(M%) * XY(I1%)
1780 M% = M% + 1
1790 FOR I2% = I1% TO D%
1800 XN(I%) = XN(I%) + A(M%) * XY(I1%) * XY(I2%)
1810 M% = M% + 1
1820 IF O% = 2 THEN GOTO 1970
1830 FOR I3% = I2% TO D%
1840 XN(I%) = XN(I%) + A(M%) * XY(I1%) * XY(I2%) * XY(I3%)
1850 M% = M% + 1
1860 IF O% = 3 THEN GOTO 1960
1870 FOR I4% = I3% TO D%
1880 XN(I%) = XN(I%) + A(M%) * XY(I1%) * XY(I2%) * XY(I3%) * XY(I4%)
1890 M% = M% + 1
1900 IF O% = 4 THEN GOTO 1950
1910 FOR I5% = I4% TO D%
1920 XN(I%) = XN(I%) + A(M%) * XY(I1%) * XY(I2%) * XY(I3%) * XY(I4%) * XY(I5%)
1930 M% = M% + 1
1940 NEXT I5%
1950 NEXT I4%
1960 NEXT I3%
1970 NEXT I2%
1980 NEXT I1%
2000 NEXT I%
2010 M% = M% - 1: XNEW = XN(1): YNEW = XN(2)

The program PROG08 could have been written more compactly but it is done this way to simplify its extension to even higher dimensions. Examples of attractors produced by this program are shown in Figures 3-18 through 3-41.

FIRPGVTF

Figure 3-18. Two-Dimensional Cubic Map

FISHMQCH

Figure 3-19. Two-Dimensional Cubic Map

FJYCBMNF

Figure 3-20. Two-Dimensional Cubic Map

FLGROKJF

Figure 3-21. Two-Dimensional Cubic Map

FMGGNDPH

Figure 3-22. Two-Dimensional Cubic Map

FNHZBEET

Figure 3-23. Two-Dimensional Cubic Map

FNUYLCUR

Figure 3-24. Two-Dimensional Cubic Map

FOVFKWKE

Figure 3-25. Two-Dimensional Cubic Map

GFUXRRRU

Figure 3-26. Two-Dimensional Quartic Map

GGNXVYVA

Figure 3-27. Two-Dimensional Quartic Map

GLURFSRH

Figure 3-28. Two-Dimensional Quartic Map

GPFMQPPB

Figure 3-29. Two-Dimensional Quartic Map

GQDIDSBT

Figure 3-30. Two-Dimensional Quartic Map

GRMJQBCS

Figure 3-31. Two-Dimensional Quartic Map

GTPMJKFS

Figure 3-32. Two-Dimensional Quartic Map

GUETJGII

Figure 3-33. Two-Dimensional Quartic Map

HGEQGOYI

Figure 3-34. Two-Dimensional Quintic Map

HHVOIEGI

Figure 3-35. Two-Dimensional Quintic Map

HMSMTNCO

Figure 3-36. Two-Dimensional Quintic Map

HQBKSKIX

Figure 3-37. Two-Dimensional Quintic Map

HQDHFCND

Figure 3-38. Two-Dimensional Quintic Map

HSARYDPN

Figure 3-39. Two-Dimensional Quintic Map

HVHDXLMS

Figure 3-40. Two-Dimensional Quintic Map

HVNTBSGW

Figure 3-41. Two-Dimensional Quintic Map

Perhaps this is a good point to pause and reiterate in what sense these objects are attractors. If you choose initial values of X and Y somewhere near the attractor within its basin of attraction and substitute these values into the equations that describe the attractor the new values of X and Y will represent a point in the plane that is closer to the attractor. After a number of iterations the point will work its way to the attractor and thereafter it will move around on the attractor in some complicated manner eventually visiting every part of the attractor. The next position can always be simply and accurately predicted from the current position but the small inevitable uncertainty in position continually increases so that a long-term prediction is impossible except to say that the point will be somewhere on the attractor. You can think of the attractor as the set of all possible long-term solutions of the equations that produced it.

Besides the error in knowing perfectly the initial conditions there are also computer round-off errors at each iteration. Given the extreme sensitivity to small errors you may wonder whether any computer is capable of calculating correctly such a chaotic process. It is true that if the same chaotic equations are iterated on two computers using different precision or round-off methods the sequence of iterates will almost certainly be completely different after a few iterations. However the general appearance of the attractor will probably be the same. In such a case we say that the solution is structurally stable or robust. Furthermore according to the shadowing lemma an appropriate small change in initial conditions will produce a chaotic sequence that follows arbitrarily close to the computed one.

Since computers always round the results of calculations to a finite number of digits (or more precisely bits) there is a limited number of allowed values. Thus successive iteration of a map will always eventually repeat a previously obtained value whereupon the solution will reproduce exactly the same sequence of states as it did before. Strictly speaking every such solution is periodic and true chaos cannot be observed with a computer. However with double precision floating point variables which are normally 64 bits there are 264 or about 1019 possible values. Until the number of iterations approaches this value there is little cause to worry. For maps higher than one-dimensional this problem is even less serious since all the variables have to reach a previously existing state at the same time.

It is also interesting to realize that infinitely many periodic solutions are embedded in each attractor. These solutions are called periodic orbits. From wherever you start on the attractor you will eventually return to a point arbitrarily close to the starting point. This result is called the Poincaré recurrence theorem after Jules-Henri Poincaré a French mathematician who a hundred years ago portended the modern era of chaos. Thus by making only a small change in the starting point it is possible in principle to return exactly to the starting point which implies a periodic orbit with a period equal to the number of iterations required to return. Most of these orbits have very large periods however.

Every point on the attractor is arbitrarily close to such a periodic orbit but the chance that a randomly chosen point on the attractor lies on such an orbit is infinitesimal. We say that the periodic orbits are dense on the attractor. These orbits though infinite in number constitute a Cantor set of measure zero. The periodic orbits are unstable in the sense that if you get just slightly off the orbit you will continue to get farther away with each iteration.

The strange attractors exhibited in this book are examples of orbital fractals. They should be distinguished from escape-time fractals which show the basin of attraction and typically display with color the number of iterations required for points outside the basin to escape beyond some pre-defined region. The Mandelbrot and Julia sets are perhaps the best-known escape-time fractals. Escape-time fractals require much longer computing times to develop but provide dazzling displays with exotic fine-scale structure.

3.6 Strange Attractor Planets

The previous figures have obvious beauty but they generally lack symmetry. Nature mixes symmetry with disorder and our sense of beauty has developed accordingly. The Earth viewed from outer space is beautiful in part because the irregular features of the clouds and continents are superimposed on a nearly perfect sphere.

There are many ways to do the same with our attractors. Suppose for example X and Y are not the horizontal and vertical position in a plane but rather the longitude and latitude on the surface of the Earth. The result is an object that might resemble a strange planet with swirling clouds oceans canals craters and other features.

Note that mapping a plane onto a sphere is a nonlinear transformation. You can't wrap a piece of paper around a globe without a large non-uniform stretching. That's why Greenland looks larger than South America on most flat maps. When a sphere is projected onto a flat computer screen or onto the page of a book it is stretched so as to magnify the central portion of the attractor and compress the edges.

If q is the longitude (measured from zero at the right edge) and f is the latitude (measured from zero at the top) the X and Y coordinates of the projection of a sphere onto the screen are given by

Xp = cos q sin f

(Eq. 3G)

Yp = cos f

We get q from X by a scaling that keeps q in the range of 0 to p radians (180 degrees) since there is no need to plot points that lie on the back side of the planet. Similarly we get f from Y by a scaling that keeps f in the range of 0 (at the North Pole) to p radians (at the South Pole). The program modifications required to accomplish this transformation are given in the listing PROG09. The program allows you to toggle back and forth between the two types of projection by pressing the P key.

PROG09 Changes Required in PROG08 to Project Attractor onto a Sphere

1000 REM TWO-D MAP SEARCH (Projected onto a Sphere)

1110 PJT% = 1                   'Projection is spherical
2260    IF PJT% = 1 THEN GOSUB 4100     'Project onto a sphere
3200 XA = (XL + XH) / 2: YA = (YL + YH) / 2
3310 IF PJT% <> 1 THEN LINE (XL
YL)-(XH
YH)

B
3320 IF PJT% = 1 THEN CIRCLE (XA
YA)
.36 * (XH - XL)
3330 TT = 3.1416 / (XMAX - XMIN): PT = 3.1416 / (YMAX - YMIN)
3750 IF Q$ = "P" THEN PJT% = (PJT% + 1) MOD 2: T% = 3: IF N > 999 THEN N = 999
4100 REM Project onto a sphere
4110 TH = TT * (XMAX - XP)
4120 PH = PT * (YMAX - YP)
4130 XP = XA + .36 * (XH - XL) * COS(TH) * SIN(PH)
4140 YP = YA + .5 * (YH - YL) * COS(PH)
4150 RETURN

Figures 3-42 through 3-57 show some examples of two-dimensional attractors projected onto a sphere. Note that the features on the attractors tend to converge at the poles at the top and bottom of the figures. This convergence could be suppressed by using an area-preserving transformation that stretches the Y-values near the poles by the same factor that the X-values are compressed. This simplest way to produce this effect is just to delete line 4140.

ECSRKVVQ

Figure 3-42. Two-Dimensional Quadratic Map Projected onto a Sphere

ECVQKGHQ

Figure 3-43. Two-Dimensional Quadratic Map Projected onto a Sphere

EKPNERVO

Figure 3-44. Two-Dimensional Quadratic Map Projected onto a Sphere

EUWACXDQ

Figure 3-45. Two-Dimensional Quadratic Map Projected onto a Sphere

FKAWYMKA

Figure 3-46. Two-Dimensional Cubic Map Projected onto a Sphere

FLQOBBRS

Figure 3-47. Two-Dimensional Cubic Map Projected onto a Sphere

FLUCBPVB

Figure 3-48. Two-Dimensional Cubic Map Projected onto a Sphere

FMEGVTLM

Figure 3-49. Two-Dimensional Cubic Map Projected onto a Sphere

GJCQPYDV

Figure 3-50. Two-Dimensional Quartic Map Projected onto a Sphere

GLQGRLUF

Figure 3-51. Two-Dimensional Quartic Map Projected onto a Sphere

GNWCAVJO

Figure 3-52. Two-Dimensional Quartic Map Projected onto a Sphere

GTNSTDPQ

Figure 3-53. Two-Dimensional Quartic Map Projected onto a Sphere

HEAUYOII

Figure 3-54. Two-Dimensional Quintic Map Projected onto a Sphere

HFJFWKAS

Figure 3-55. Two-Dimensional Quintic Map Projected onto a Sphere

HLTQSSRK

Figure 3-56. Two-Dimensional Quintic Map Projected onto a Sphere

HOKEFWLH

Figure 3-57. Two-Dimensional Quintic Map Projected onto a Sphere

If you are using PowerBASIC or its predecessor TurboBASIC and VGA graphics you will note a slight incompatibility with the CIRCLE command that causes the size of the circle that surrounds the attractor to vary from case to case. In these dialects of BASIC the radius of the circle in SCREEN modes 11 and 12 is specified in units of the screen height rather than its width. If you encounter this problem try replacing the .36 * (XH - XL) in line 3320 with .5 * (YH - YL).

Planes and spheres are not the only two-dimensional surfaces onto which attractors can be projected. A cylinder is another possibility. The cylinder can be oriented with its axis either horizontal or vertical or tilted at some arbitrary angle. A torus is another possibility. You may be able to think of other more exotic surfaces onto which the attractors can be projected.

3.7 Designer Plaids

It is interesting that all the one-dimensional maps described in the previous chapter are included in the two-dimensional cases. One needs only to set the coefficients of the Y-equation to zero. For example a two-dimensional map equivalent to the logistic equation is given by the code EMu%M9. However since Y doesn't change with successive iterations a graph of Y versus X is simply a straight horizontal line.

To display the logistic parabola we need to replace X with the next iterate of X and Y with the second next iterate of X. Two successive iterations of a quadratic map requires a fourth-order equation. A code that will accomplish this is GMu%M13NHUIM10.

There are other examples of two-dimensional maps that are really one-dimensional maps in disguise. Suppose Xn+1 depends only on Yn and Yn+1 depends only on Xn. Then Xn+2 will depend only on Xn and we have a one-dimensional map for X in which Y is merely an intermediate value of X. The most general fifth-order polynomial example of such a case is

Xn+1 = a1 + a17Yn + a18Yn2 + a19Yn3 + a20Yn4 + a21Yn5

(Eq. 3H)

Yn+1 = a22 + a23Xn + a24Xn2 + a25Xn3 + a26Xn4 + a27Xn5

This case can be achieved by setting the remaining thirty coefficients to zero in PROG09 by adding the following line after line 2730:

2735 IF (I% > 1 AND I% < M% / 2 - O%) OR I% > M% / 2 + O% + 1 THEN MID$(CODE$ I% + 1 1) = "M"

The result is to produce a 25th-order one-dimensional polynomial map displayed in two dimensions.

Figures 3-58 through 3-62 show sample attractors obtained in this way. Notice that they fill in rectangular regions resembling a plaid tartan in sharp contrast to all the previous cases. These attractors are especially appropriate for projecting onto a sphere since the features line up east-west along parallels and north-south along meridians. Figures 3-63 and 3-64 show some examples of plaid planetary attractors.

ECMMMEWH

Figure 3-58. Two-Dimensional Quadratic Plaid Map

FNMMMMMM

Figure 3-59. Two-Dimensional Cubic Plaid Map

GEMMMMMM

Figure 3-60. Two-Dimensional Quartic Plaid Map

HKMMMMMM

Figure 3-61. Two-Dimensional Quintic Plaid Map

ERMMMQEA

Figure 3-62. Two-Dimensional Quadratic Plaid Map on a Sphere

HOMMMMMM

Figure 3-63. Two-Dimensional Quintic Plaid Map on a Sphere

You might wish to try adding colors to emulate a decorative cloth pattern. On way to do this is to color the pixels according to the number of times they are visited by the orbit. This is easily done by changing line 2300 in the program to

2300 PSET (XP YP) (POINT (XP YP) + 1) MOD 16

which causes the color of the existing point at (XP YP) to be tested and then plotted with the next color in the palette of 16 colors. In the next chapter we will discuss other ways to produce colorful attractors.

3.8 Strange Attractors that Don't

From the foregoing discussion you might conclude that all chaotic equations produce strange attractors. Such is not the case. Under certain conditions the successive iterates of an equation will wander chaotically throughout a region of the plane. There is no basin of attraction and initial conditions near but outside the chaotic region are not drawn to the region but rather lie on closed curves. Although the chaotic region is not a strange attractor it may have considerable beauty.

For a chaotic solution not to attract the area occupied by a cluster of nearby initial conditions must remain the same with successive iterations. The cluster will generally contract in one direction and expand in the other but the contraction and expansion just cancel producing a long thin filament of constant area. A characteristic of such a case is that the two Lyapunov exponents are equal in magnitude but of opposite signs. Such a system is said to be area-preserving or Hamiltonian.

You might think that Hamiltonian systems are relatively rare in nature since they require a special condition. However there are many important examples of Hamiltonian chaos. They arise because there are quantities in nature such as energy and angular momentum that in the absence of friction remain accurately constant no matter how complicated the behavior of the system. We say these quantities are conserved or that they are constants of the motion. The motion of a planet orbiting a binary-star system or the motion of an electron near the null in a magnetic field exhibit Hamiltonian chaos. A more familiar example is the filamentation of milk when it is stirred into coffee in which case the volume of the milk is conserved because liquids are nearly incompressible.

With equations such as those we have been using with randomly chosen coefficients the chance of inadvertently finding an area-preserving solution is essentially zero. However by placing appropriate conditions on the coefficients we can guarantee such solutions. An example of an area-preserving two-dimensional polynomial map is the following:

Xn+1 = a1 + a2Xn + a3Xn2 + a4Xn3 + a5Xn4 + a6Xn5 ± Yn

(Eq. 3I)

Yn+1 = a22 ± Xn

This map is fifth-order to provide seven arbitrary coefficients that ensure a large number of solutions. The coefficient labels are consistent with the general two-dimensional fifth-order map in which 33 of the coefficients have been set to zero. The two terms preceded with ± (a17 and a23) have coefficients of either +1 or -1 and this feature guarantees an area-preserving solution. If the signs are the same (both plus or both minus) chaotic solutions are not found. Hamiltonian chaos can occur when the signs are opposite. The negative product of these two coefficients is the Jacobian of the map (J = -a17a23). The Jacobian is a measure of the net contraction and it must equal 1.0 for a Hamiltonian system.

Hamiltonian cases can be produced by adding to PROG09 after line 2730 the following lines:

2735 IF I% > O% + 1 AND I% <> M% / 2 + 1 THEN MID$(CODE$ I% + 1 1) = "M"

2736 MID$(CODE$ M% / 2 - O% + 2 1) = "W": MID$(CODE$ M% / 2 + 3 1) = "C"

Sample Hamiltonian chaotic solutions are shown in Figures 3-64 through 3-71. Most of the cases resemble chains of islands in which each island contains a fixed point surrounded by closed contours that are not shown. These cases were produced using initial values of X = Y = 0.05. Other initial conditions would produce completely different pictures since there is no attractor.

HDCEEYSM

Figure 3-64. Two-Dimensional Quintic Hamiltonian Map

HFOWFIPM

Figure 3-65. Two-Dimensional Quintic Hamiltonian Map

HHJJYPHM

Figure 3-66. Two-Dimensional Quintic Hamiltonian Map

HIOMYNHM

Figure 3-67. Two-Dimensional Quintic Hamiltonian Map

HROXTCGM

Figure 3-68. Two-Dimensional Quintic Hamiltonian Map

HVEMBUMM

Figure 3-69. Two-Dimensional Quintic Hamiltonian Map

HVFMTTCM

Figure 3-70. Two-Dimensional Quintic Hamiltonian Map

HVNSGEQM

Figure 3-71. Two-Dimensional Quintic Hamiltonian Map

Hamiltonian cases have a different look from non-Hamiltonian strange attractors. The difference is even more pronounced if you watch while they develop on the computer screen. Whereas the regions of a strange attractor tend to be visited uniformly and apparently randomly the Hamiltonian cases develop much more slowly. The points often wander over a small region for tens of thousands of iterations and then they will suddenly begin to fill in a new distinct region that had never been visited before. Consequently many more iterations are required to determine the stability and chaotic nature of the solution. You need to be patient while the computer calculates.

The different time behavior of these cases raises an important issue. When you view any of the figures in this book you are seeing a static object. However it was produced by a dynamic process. Information about the sequence in which the points accumulated has been lost. This additional information is recovered when you watch the attractors develop on your computer screen. Most of the attractors fill in uniformly. The contrast gets progressively greater much as a photographic print being developed.

However the Hamiltonian cases develop more slowly and in stages. If your computer has a color monitor you might try exhibiting this sequence by plotting the points in color and changing the color every few thousand iterations. Some examples using this technique will be shown in Section 7.5. If you try this for the non-Hamiltonian attractors the colors overlap and merge into a uniform gray or you just see the most recent color. For the Hamiltonian cases beautiful color patterns can be produced. Otherwise continue on to the next chapter where various color techniques will be discussed.

3.9 A New Dimension in Sound

With one-dimensional maps we tried to make music by letting successive iterates control the pitch of the musical notes all of which were of the same duration. The same procedure can be used with two-dimensional maps. However we have a second variable at our disposal and so let's use it to control the duration of each note. With actual music it turns out that there are many more notes of short duration than of long duration. There are roughly twice as many half notes as whole notes and twice as many quarter notes as half notes and so forth. This remarkable result seems to hold for all types of music from different composers and cultures. It is evidence of hidden determinism in music.

The program modification PROG10 uses the X-value to control the pitch and the Y-value to control the duration of the notes. For convenience we assume that the longest note is a whole note and the shortest note is a sixteenth note. Dotted notes and rests are not allowed.

The listing PROG10 also adds to the program a menu screen that reminds you of the S command which toggles the sound on and off and the P command which toggles the projection between planar and spherical. We also introduce an A command to initiate the search for attractors a D command to toggle between one-dimensional and two-dimensional maps an I command to let you input the code of an attractor that you know and an X command to exit the program. Pressing any other key will display the menu screen.

PROG10 Changes Required in PROG09 to Produce Chaotic Music and Provide a Menu Screen

1000 REM TWO-D MAP SEARCH (With Music and Menu Screen)

1100 SND% = 1                   'Turn sound on
1110 PJT% = 0                   'Projection is planar
1170 GOSUB 4200                 'Display menu screen
1180 IF Q$ = "X" THEN GOTO 1250 'Exit immediately on command
2450 IF QM% > 0 THEN GOTO 2490  'Skip tests when not in search mode
2640 IF QM% > 0 THEN GOTO 2730  'Not in search mode
2650    O% = 2 + INT((OMAX% - 1) * RND)
2660    CODE$ = CHR$(59 + 4 * D% + O%)
2680    GOSUB 4700              'Get value of M%
3530 IF D% > 1 THEN DUR = 2 ^ INT(.5 * (YH - YL) / (YNEW - 9 * YL / 8 + YH / 8))
3610 IF ASC(Q$) > 96 THEN Q$ = CHR$(ASC(Q$) - 32)   'Convert to upper case
3630 IF Q$ = "" OR INSTR("ADIPSX"
Q$) = 0 THEN GOSUB 4200
3640 IF Q$ = "A" THEN T% = 1: QM% = 0
3680 IF Q$ = "D" THEN D% = 1 + (D% MOD 2): T% = 1
3730 IF Q$ = "I" THEN IF T% <> 1 THEN SCREEN 0: WIDTH 80: COLOR 15
1: CLS : LINE INPUT "Code? "; CODE$: IF CODE$ = "" THEN Q$ = " ": CLS :  ELSE T% = 1: QM% = 1: GOSUB 4700
3790 IF Q$ = "X" THEN T% = 0
4200 REM Display menu screen
4210 SCREEN 0: WIDTH 80: COLOR 15
1: CLS
4220 WHILE Q$ = "" OR INSTR("AIX"
Q$) = 0
4230    LOCATE 1
27: PRINT "STRANGE ATTRACTOR PROGRAM"
4260    PRINT : PRINT
4270    PRINT TAB(27); "A: Search for attractors"
4300    PRINT TAB(27); "D: System is"; STR$(D%); "-D polynomial map"
4370    PRINT TAB(27); "I: Input code from keyboard"
4400    PRINT TAB(27); "P: Projection is ";
4410        IF PJT% = 0 THEN PRINT "planar   "
4420        IF PJT% = 1 THEN PRINT "spherical"
4540    PRINT TAB(27); "S: Sound is ";
4550        IF SND% = 0 THEN PRINT "off"
4560        IF SND% = 1 THEN PRINT "on "
4600    PRINT TAB(27); "X: Exit program"
4610    Q$ = INKEY$
4620    IF Q$ <> "" THEN GOSUB 3600     'Respond to user command
4630 WEND
4640 RETURN
4700 REM Get dimension and order
4710 D% = 1 + INT((ASC(LEFT$(CODE$
1)) - 65) / 4)
4740 O% = 2 + (ASC(LEFT$(CODE$
1)) - 65) MOD 4
4750 M% = 1: FOR I% = 1 TO D%: M% = M% * (O% + I%): NEXT I%
4770 IF LEN(CODE$) = M% + 1 OR QM% <> 1 THEN GOTO 4810
4780    BEEP        'Illegal code warning
4790    WHILE LEN(CODE$) < M% + 1: CODE$ = CODE$ + "M": WEND
4800    IF LEN(CODE$) > M% + 1 THEN CODE$ = LEFT$(CODE$
M% + 1)
4810 RETURN

As you listen to the music produced by the various attractors you may discover relations between the quality of the music and the appearance of the attractor. The cases that seem most musical tend to have certain visual characteristics which will be left for you to discover. Do attractors that appeal to the eye also appeal to the ear?

After you have generated some music of your own you may wish to try some of the cases in Table 3.1 using the I command to input them to the program. They have been selected for their musical quality and are limited to quadratic maps to simplify typing their codes. An interesting study would be to accumulate your own longer list of musical attractors and to see if they preferentially have certain fractal dimensions and Lyapunov exponents. If so then it should be possible to program the computer to be a music critic as well as an art critic.

Table 3.1 List of Some Musical Attractors and their Characteristics

After listening to the enormous variety of musical sequences that can be generated by this technique you might wonder whether your favorite musical composition could be compressed into a short code and generated using iterated maps. After all even for the simple cases in the table above there are about 6 x 1016 different codes and each code corresponds to a different piece of music.

However a typical musical piece might have hundreds or thousands of notes each of which can represent dozens of pitches and many durations. Thus we can be fairly confident using the principles of information theory that such extreme compression is unlikely unless there is considerably more structure to music than is apparent. However if you only want to generate a short tune with a few notes there might well be a way to do so using this technique. If you are mathematically inclined take it as a challenge to find a way to do this.

The generation of computer music using chaotic iterated maps is a promising technique still in its infancy. You may wish to incorporate more sophisticated musical rules into the program to generate music that is much more pleasing than what results from this simple procedure. Furthermore an interesting project would be to turn the process around and see if music written by humans resembles a strange attractor and if so to measure its fractal dimension and Lyapunov exponent. Perhaps music of different types or by different composers would have different values of these quantities.

CHAPTER 4

Attractors of Depth

A two-dimensional world is a mere shadow of reality. The techniques described in the previous chapters are easily extended to produce attractors embedded in the three-dimensional space in which we live. The challenge is in finding ways to exhibit and visualize such three-dimensional objects within the limitations of the computer screen and printed page. This chapter will emphasize new visualization techniques and will provide many new examples of strange attractors that have depth as well as width and height.

4.1 Projections

The procedure for seeking attractors in three dimensions (which we might whimsically call strange attractors of the third kind) is just like the two-dimensional case except that we introduce a third variable Z to accompany X and Y. You can think of Z as representing the position in a direction out of the screen or page on which the attractor is displayed. We will take the direction of positive Z to be out of the page and negative Z to be behind the page as is customary for a conventional right-handed coordinate system. The term right-handed comes from the convention that if you point the fingers of your right hand in the direction of the X-axis and curl them so that they point along the Y-axis your thumb will point in the Z-direction. This choice is purely arbitrary but is widely accepted.

The simplest system of equations that produces strange attractors embedded in a three-dimensional space is a set of coupled quadratic equations the most general form of which is given by

Xn+1 = a1 + a2Xn + a3Xn2 + a4XnYn + a5XnZn + a6Yn

+ a7Yn2 + a8YnZn + a9Zn + a10Zn2

(Eq. 4A)

Yn+1 = a11 + a12Xn + a13Xn2 + a14XnYn + a15XnZn + a16Yn

+ a17Yn2 + a18YnZn + a19Zn + a20Zn2

Zn+1 = a21 + a22Xn + a23Xn2 + a24XnYn + a25XnZn + a26Yn

+ a27Yn2 + a28YnZn + a29Zn + a30Zn2

These equations have thirty coefficients which allow an enormous variety of attractors. The extension to equations with order higher than two is straightforward. Three-dimensional cubic equations have sixty coefficients quartic equations have 105 coefficients and quintic equations have 168 coefficients. The number of coefficients for order O is given by (O + 1)(O + 2)(O + 3) / 2. We will code the second-order through fifth-order systems in three dimensions with the initial letters I J K and L respectively.

Note that 168 coefficients allows 25168 or about 10234 combinations. This is a truly astronomical number. Even if only a small fraction of them correspond to distinct strange attractors their number enormously exceeds the number of electrons protons and neutrons in the entire universe! This number is a mere 1079. Thus the number of fifth-order three-dimensional strange attractors is essentially infinite. You can have a large collection of your own none of which are likely to be reproduced by anyone else unless you give them the code you used to produce them. The code is like a combination lock with 168 settings that must all be entered correctly and in the proper order.

Now we must confront the issue of how best to display an object composed of points in a three-dimensional space. Such problems are in the domain of a new specialty called visualization which we may define as the use of computer imagery to gain insight into complex phenomena. The need for improved visualization techniques has emerged from the rapidly growing use of computers as the primary tool for scientific calculation and modeling. As computers become more powerful it is increasingly important to devise methods of dealing with large quantities of data. The eye and brain are very efficient at discerning visual patterns and these patterns permit an intuitive understanding of complicated processes in a way that equations often cannot. Scientists have recently developed impressive visualization techniques simple versions of which are presented here.

The simplest method is to ignore one of the coordinates and to plot the points in the remaining two dimensions. This method is equivalent to looking at the shadow cast by the object when illuminated from directly above by a point-source of light a large distance away. If the light source is on the Z-axis we say the attractor is projected onto the XY-plane. The screen used in conjunction with a slide projector is such a plane. Of course considerable information about the attractor is lost in such a projection but the method is a convenient starting point and it is simple to program.

The listing PROG11 provides the changes that must be made in the program PROG10 to extend the attractor search to three dimensions with order up to five. Since the search slows down considerably in three dimensions with such a large number of coefficients especially if you don't have a compiled version of BASIC and a fast computer the program saves for each case found the code fractal dimension and Lyapunov exponent in a disk file with the name SA.DIC (Strange Attractor Dictionary). This feature allows you to run the program unattended and to collect the attractors it finds. We will later modify the program to let you examine the cases that you collect.

PROG11 Changes Required in PROG10 to Search for Strange Attractors in Three Dimensions

1000 REM THREE-D MAP SEARCH

1020 DIM XS(499)
YS(499)
ZS(499)
A(504)
V(99)
XY(4)
XN(4)
1070 D% = 3                     'Dimension of system
1100 SND% = 0                   'Turn sound off
1530 Z = .05
1550 XE = X + .000001: YE = Y: ZE = Z
1600 ZMIN = XMIN: ZMAX = XMAX
1720 M% = 1: XY(1) = X: XY(2) = Y: XY(3) = Z
2010 M% = M% - 1: XNEW = XN(1): YNEW = XN(2): ZNEW = XN(3)
2160    IF Z < ZMIN THEN ZMIN = Z
2170    IF Z > ZMAX THEN ZMAX = Z
2210 XS(P%) = X: YS(P%) = Y: ZS(P%) = Z
2410 IF ABS(XNEW) + ABS(YNEW) + ABS(ZNEW) > 1000000! THEN T% = 2
2460 IF N >= NMAX THEN T% = 2: GOSUB 4900   'Strange attractor found
2470 IF ABS(XNEW - X) + ABS(YNEW - Y) + ABS(ZNEW - Z) < .000001 THEN T% = 2
2530 Z = ZNEW
2910 XSAVE = XNEW: YSAVE = YNEW: ZSAVE = ZNEW
2920 X = XE: Y = YE: Z = ZE: N = N - 1
2950 DLZ = ZNEW - ZSAVE
2960 DL2 = DLX * DLX + DLY * DLY + DLZ * DLZ
3010    ZE = ZSAVE + RS * (ZNEW - ZSAVE)
3020    XNEW = XSAVE: YNEW = YSAVE: ZNEW = ZSAVE
3140 IF ZMAX - ZMIN < .000001 THEN ZMIN = ZMIN - .0000005: ZMAX = ZMAX + .0000005
3400 LOCATE 1
1: IF LEN(CODE$) < 62 THEN PRINT CODE$
3410 IF LEN(CODE$) >= 62 THEN PRINT LEFT$(CODE$
57) + "..."
3680 IF Q$ = "D" THEN D% = 1 + (D% MOD 3): T% = 1
3920 IF N = 1000 THEN D2MAX = (XMAX - XMIN) ^ 2 + (YMAX - YMIN) ^ 2 + (ZMAX - ZMIN) ^ 2
3940 DX = XNEW - XS(J%): DY = YNEW - YS(J%): DZ = ZNEW - ZS(J%)
3950 D2 = DX * DX + DY * DY + DZ * DZ
4760 IF D% = 3 THEN M% = M% / 2
4900 REM Save attractor to disk file SA.DIC
4910 OPEN "SA.DIC" FOR APPEND AS #1
4920 PRINT #1
CODE$; : PRINT #1
USING "##.##"; F; L
4930 CLOSE #1
4940 RETURN

Some examples of the attractors produced by PROG11 are shown in Figures 4-1 through 4-16. Note that the fractal dimension shown for each case is the dimension of the actual attractor and not the dimension of its projection. Thus the fractal dimension can be as large as 3 even though the projection has dimension of at most 2. The projection of a point (zero dimensions) onto a surface is a point the projection of a line (one dimension) is a line the projection of a surface (two dimensions) is a surface but the projection of a solid (three dimensions) onto a surface is only a surface (two dimensions).

IJKRADSX

Figure 4-1. Projection of Three-Dimensional Quadratic Map

ILURCEGO

Figure 4-2. Projection of Three-Dimensional Quadratic Map

IMTISVBK

Figure 4-3. Projection of Three-Dimensional Quadratic Map

INRRXLCE

Figure 4-4. Projection of Three-Dimensional Quadratic Map

IOHGWFIH

Figure 4-5. Projection of Three-Dimensional Quadratic Map

IOLORGSF

Figure 4-6. Projection of Three-Dimensional Quadratic Map

IQWGBEJQ

Figure 4-7. Projection of Three-Dimensional Quadratic Map

IWDWOGDG

Figure 4-8. Projection of Three-Dimensional Quadratic Map

JJXLHCRX

Figure 4-9. Projection of Three-Dimensional Cubic Map

JKWNBMTO

Figure 4-10. Projection of Three-Dimensional Cubic Map

JLRTXANM

Figure 4-11. Projection of Three-Dimensional Cubic Map

KOCWVUCR

Figure 4-12. Projection of Three-Dimensional Quartic Map

KPGQHOXC

Figure 4-13. Projection of Three-Dimensional Quartic Map

KYECWCLF

Figure 4-14. Projection of Three-Dimensional Quartic Map

LMLMBFUT

Figure 4-15. Projection of Three-Dimensional Quintic Map

LPJDQLOH

Figure 4-16. Projection of Three-Dimensional Quintic Map

On the whole attractors in three dimensions projected onto a plane are not particularly different or better than the two-dimensional examples of the previous chapter. Ones with high fractal dimensions (near and above 2) tend to be uninteresting when projected onto two dimensions because they are too filled-in. Note also that all the two-dimensional cases are included as special cases of the three-dimensional ones and that they can be recovered by setting the appropriate coefficients to zero. For example the Hénon map can be reproduced in three dimensions using the code IWM?M2PM5WM18. You may wish to try entering this case into the program using the I command. Be sure to count the number of M's very carefully and to use capital letters.

The attractors displayed in the previous figures are projected onto the XY-plane. They could equally well be projected onto the YZ or ZX-plane. With a bit more effort it would be possible to project them onto a plane inclined at an arbitrary angle. Attractors are most visually appealing when viewed from a particular direction. The formulas that transform a point with coordinates (X Y Z) into a two-dimensional projection (Xp Yp) with viewing angles (q f) in spherical coordinates are

Xp = - X sin q + Y cos q

(Eq. 4B)

Yp = - X sin q cos f - Y sin q cos f + Z sin f

With a sufficiently powerful computer you could rotate the attractor to produce an animated display. You may wish to experiment with these ideas.

There are a number of ways to modify the program to change the orientation of the projection. The simplest (though not very practical) way is just to wait until the search turns up the same attractor viewed from a different angle. Note in Equation 4A for example that if you interchange the coefficients in an appropriate way the result is to replace X with Y Y with Z and Z with X. If the attractor code were IABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^ the code IKPQRNSTOLMUZ[\X]^YVWAFGHDIJEBC would produce a projection of the same attractor onto the YZ-plane and the code IU]^Y\VWXZ[AIJEHBCDFGKSTORLMNPQ would produce a projection of the attractor onto the ZX-plane. Figures 4-17 and 4-18 show the result of applying these transformations to the attractor in Figure 4-4.

IHSTPPHH

Figure 4-17. Attractor in Figure 4-4 Projected onto the YZ-Plane

IJNMYBMY

Figure 4-18. Attractor in Figure 4-4 Projected onto the ZX-Plane

4.2 Shadows

In the previous figures the attractors were projected as if illuminated from directly along the line of sight. Now suppose the point source of illumination is moved slightly off to one side and that you observe the attractor against a background screen. Each point making up the attractor appears as a dot of reflected light above the background plane and produces a shadow point opposite the illumination a distance proportional to the position of the point above the plane. For this purpose we assume the most distant point of the attractor is touching the screen and the nearest point is out of the screen a distance equal to its width. We'll assume it is illuminated from above your left shoulder so that the shadow is below and to the right in keeping with the Microsoft style guidelines as exhibited in recent versions of Windows.

To produce a shadow we need a background shade of gray intermediate between the black and white that we have been using so far. If your computer has at least EGA graphics this poses no difficulty. There are two grays COLOR 8 which is 25% illuminated and COLOR 7 which is 75% illuminated. We'll use COLOR 8 which is the darker of the two. For convenience Table 4.1 lists the sixteen default colors provided with SCREEN modes 7 through 13.

Table 4.1 Default EGA and VGA colors for SCREEN modes 7 to 13

If you have CGA graphics you might try plotting the points in white (COLOR 3) and their shadow in black (COLOR 0) on a magenta (COLOR 2) background using SCREEN 1 (320 by 200 resolution) and PALETTE 1. In any case it may help to adjust the intensity control on the monitor for an easily visible shadow.

Since we have another shade of gray available we can use it to control the brightness of the points plotted. The first time a screen pixel is illuminated by a point on the attractor we will use low intensity white (COLOR 7) and subsequent times we will use high intensity white (COLOR 15). If your computer has a monochrome graphics monitor that maps the other colors to various shades of gray you can extend this technique to provide additional gray levels producing an attractor whose brightness corresponds to the frequency its various regions are visited. This trick helps to compensate for the limited spatial resolution of the computer screen.

It also helps to draw a grid on the background to make it more obvious that the attractor is sitting above the screen. The grid is drawn in black (COLOR 0) the same as the shadow.

If your computer has at least EGA capability the program PROG12 will produce the desired shadow display. It allows you to toggle between projections and shadows by pressing the R key.

PROG12 Changes Required in PROG11 to Display Shadows

1000 REM THREE-D MAP SEARCH (With Shadow Display)

1020 DIM XS(499)
YS(499)
ZS(499)
A(504)
V(99)
XY(4)
XN(4)
COLR%(15)
1120 TRD% = 1                   'Display third dimension as shadow
1370 GOSUB 5600                 'Set colors
2300    GOSUB 5000                      'Plot point on screen
3210 IF D% < 3 THEN GOTO 3310
3230    IF TRD% = 1 THEN LINE (XL
YL)-(XH
YH)
COLR%(1)
BF: GOSUB 5400
3430 TIA = .05               'Tangent of illumination angle
3440 XZ = -TIA * (XMAX - XMIN) / (ZMAX - ZMIN)
3450 YZ = TIA * (YMAX - YMIN) / (ZMAX - ZMIN)
3630 IF Q$ = "" OR INSTR("ADIPRSX"
Q$) = 0 THEN GOSUB 4200
3760 IF Q$ = "R" THEN TRD% = (TRD% + 1) MOD 2: T% = 3: IF N > 999 THEN N = 999
4460    PRINT TAB(27); "R: Third dimension is ";
4470        IF TRD% = 0 THEN PRINT "projection"
4480        IF TRD% = 1 THEN PRINT "shadow    "
5000 REM Plot point on screen
5060 IF TRD% = 0 THEN PSET (XP
YP)
5070 IF TRD% <> 1 THEN GOTO 5130
5090    C% = POINT(XP
YP)
5100    IF C% = COLR%(2) THEN PSET (XP
YP)
COLR%(3) ELSE IF C% <> COLR%(3) THEN PSET (XP
YP)
COLR%(2)
5110    XP = XP - XZ * (Z - ZMIN): YP = YP - YZ * (Z - ZMIN)
5120    IF POINT(XP
YP) = COLR%(1) THEN PSET (XP
YP)
0
5130 RETURN
5400 REM Plot background grid
5410 FOR I% = 0 TO 15           'Draw 15 vertical grid lines
5420    XP = XMIN + I% * (XMAX - XMIN) / 15
5430    LINE (XP
YMIN)-(XP
YMAX)
0
5440 NEXT I%
5450 FOR I% = 0 TO 10           'Draw 10 horizontal grid lines
5460    YP = YMIN + I% * (YMAX - YMIN) / 10
5470    LINE (XMIN
YP)-(XMAX
YP)
0
5480 NEXT I%
5490 RETURN
5600 REM Set colors
5620 COLR%(0) = 0: COLR%(1) = 8: COLR%(2) = 7: COLR%(3) = 15
5720 RETURN

The angle of illumination is determined by the .05 in line 3430. You might try different values. The value of .05 is the tangent of both the horizontal and vertical angle that the source of illumination makes with the perpendicular to the plane. The angle is about 3 degrees toward the left and 3 degrees toward the top of the figure. Sample attractors produced by PROG12 are shown in Figures 4-19 through 4-34.

IGMNIKFC

Figure 4-19. Three-Dimensional Quadratic Map with Shadows

IGNXQDPR

Figure 4-20. Three-Dimensional Quadratic Map with Shadows

IHHFGLDK

Figure 4-21. Three-Dimensional Quadratic Map with Shadows

IHJNHTBG

Figure 4-22. Three-Dimensional Quadratic Map with Shadows

IIPPSGTM

Figure 4-23. Three-Dimensional Quadratic Map with Shadows

IKNXVQUE

Figure 4-24. Three-Dimensional Quadratic Map with Shadows

ILITWKWU

Figure 4-25. Three-Dimensional Quadratic Map with Shadows

INWJRPOX

Figure 4-26. Three-Dimensional Quadratic Map with Shadows

IPGPJYPM

Figure 4-27. Three-Dimensional Quadratic Map with Shadows

IPOBLRCR

Figure 4-28. Three-Dimensional Quadratic Map with Shadows

ISNQBNOJ

Figure 4-29. Three-Dimensional Quadratic Map with Shadows

JHYWCCFN

Figure 4-30. Three-Dimensional Cubic Map with Shadows

KIRCGTGY

Figure 4-31. Three-Dimensional Quartic Map with Shadows

KNRRVWRE

Figure 4-32. Three-Dimensional Quartic Map with Shadows

LLQEHSEA

Figure 4-33. Three-Dimensional Quintic Map with Shadows

LNMIWCGD

Figure 4-34. Three-Dimensional Quintic Map with Shadows

If you look closely at the figures with shadows you will note that it is hard to tell whether one portion of the attractor lies above or below another portion. The reason for this is that we have not allowed the closer portion of the attractor to cast a shadow on the more distant portion. To do so requires a complicated program which will be left as a challenge for you.

If you attempt to improve the shadow display in this way you will need to store in an array the largest Z-value corresponding to each screen pixel. With VGA (640 by 480) you will need 600 K bytes even if you convert the Z-values into integers. Most versions of BASIC limit the size of arrays to 64 K bytes and the disk operating system usually limits the total program size to about 600 K bytes. Thus you will probably have to use a lower screen resolution or devise a more compact coding scheme. For example if you use only 16 values of Z you can store two screen pixels per byte which is four times better than storing the Z-value of each pixel as a two-byte integer. Alternately you might store the Z-values of only those pixels that are illuminated but then you will need to devise a quick way to locate the proper element in the array corresponding to each pair of screen coordinates.

Before each point on the attractor is plotted it is necessary to be sure it doesn't fall in the shadow of a previously plotted point. If it doesn't then it can be plotted but then you have to determine whether it occludes any previously plotted point. Alternately you can first plot all the points and then scan the image starting from the side toward the illumination blocking out any points that fall in the shadow of another point.

4.3 Bands

Another way to display the third dimension is with elevation contours such as those found on topographic maps. With enough points you could plot only those that have specific values of Z. Of course the chance that a point has any particular exact value of Z is negligibly small and so the points would accumulate on the screen very slowly. To make the method work you have to plot all the points that lie within bands centered on the desired values.

You have freedom to choose the width of the bands. With narrow bands the contours resemble distinct lines but they form very slowly. With wide bands the gaps between the bands are hard to see. By making the bands half as wide as the spacing between the contours the bright and dark spaces are equal in width and they form rapidly and are easy to see.

You also need to decide how many contours to use. You need at least several to make the method work but if you use too many they begin to run together at modest screen resolution and number of iterations. For the cases shown here we use fifteen bands as a reasonable compromise.

Since we used a four-level gray scale to produce the shadows in the previous displays we will also use it here to give the bands a softer shading. Of course this requires a computer with at least EGA graphics. If your computer has CGA graphics you will only see two shades (black and white) with 30 bands in SCREEN mode 2.

The changes required in the program to produce contour bands are shown in the listing PROG13.

PROG13 Changes Required in PROG12 to Display Contour Bands

1000 REM THREE-D MAP SEARCH (With Contour Bands)

1120 TRD% = 2                   'Display third dimension as bands
3760 IF Q$ = "R" THEN TRD% = (TRD% + 1) MOD 3: T% = 3: IF N > 999 THEN N = 999
4490        IF TRD% = 2 THEN PRINT "bands     "
5130 IF TRD% = 2 THEN PSET (XP
YP)
COLR%(INT(60 * (Z - ZMIN) / (ZMAX - ZMIN) + 4) MOD 4)
5240 RETURN

Some sample attractors with contour bands produced by the program PROG13 are shown in Figures 4-35 through 4-50.

IFJLRNTK

Figure 4-35. Three-Dimensional Quadratic Map with Contour Bands

IJEKESGY

Figure 4-36. Three-Dimensional Quadratic Map with Contour Bands

IJMUYUNO

Figure 4-37. Three-Dimensional Quadratic Map with Contour Bands

IJMYRKHS

Figure 4-38. Three-Dimensional Quadratic Map with Contour Bands

INDVLLVR

Figure 4-39. Three-Dimensional Quadratic Map with Contour Bands

IQCBIKKB

Figure 4-40. Three-Dimensional Quadratic Map with Contour Bands

ISKKGLKS

Figure 4-41. Three-Dimensional Quadratic Map with Contour Bands

JJINFJIC

Figure 4-42. Three-Dimensional Cubic Map with Contour Bands

JLEDOSLN

Figure 4-43. Three-Dimensional Cubic Map with Contour Bands

JPLFQMNU

Figure 4-44. Three-Dimensional Cubic Map with Contour Bands

KKGKUQRW

Figure 4-45. Three-Dimensional Quartic Map with Contour Bands

KKLXFIMK

Figure 4-46. Three-Dimensional Quartic Map with Contour Bands

KQMPLCKL

Figure 4-47. Three-Dimensional Quartic Map with Contour Bands

LLLRGRWW

Figure 4-48. Three-Dimensional Quintic Map with Contour Bands

LOKEHGAF

Figure 4-49. Three-Dimensional Quintic Map with Contour Bands

LQGUEQNP

Figure 4-50. Three-Dimensional Quintic Map with Contour Bands

4.4 Colors

It's not hard to guess that the next logical step is to use the full array of colors available on a computer with a color monitor. In SCREEN mode 9 (EGA) and SCREEN mode 12 (VGA) sixteen colors can be displayed simultaneously from a palette of 64 (EGA) or 262 144 (VGA). SCREEN mode 13 (VGA) which is supported by some BASIC versions allows 256 colors but the screen resolution of 320 by 200 is inadequate for our purposes and thus it will not be used. In SCREEN mode 1 (CGA) only four colors can be displayed from one of two palettes. We will assume the computer has EGA or VGA capabilities but the program will also work with CGA if you use SM% = 1 in line 1030. The program is written to simplify extending the technique to future new graphics modes with more colors and higher resolution provided they are supported by your BASIC compiler.

We will convert the Z-values into fifteen different colors (COLOR 1 through COLOR 15). The sixteenth (COLOR 0) is the background color and will not be used. The default values of the colors are given in Table 4.1. The changes required to the program are shown in the listing PROG14.

PROG14 Changes Required in PROG13 to Display Colors

1000 REM THREE-D MAP SEARCH (With Color Display)

1120 TRD% = 3                   'Display third dimension in color
3760 IF Q$ = "R" THEN TRD% = (TRD% + 1) MOD 4: T% = 3: IF N > 999 THEN N = 999: GOSUB 5600
4500        IF TRD% = 3 THEN PRINT "colors    "
5160 IF TRD% = 3 THEN PSET (XP
YP)
COLR%(INT(NC% * (Z - ZMIN) / (ZMAX - ZMIN) + NC%) MOD NC%)
5610 NC% = 15                   'Number of colors
5630 IF TRD% = 3 THEN FOR I% = 0 TO NC%: COLR%(I%) = I% + 1: NEXT I%

Some sample color attractors produced by the program PROG14 are shown in Plates 1 through 8. Some of these examples are projected onto a sphere. Note that the interposition of dots of different colors in some of the figures gives the impression that there are many more than sixteen colors. The addition of color usually enhances the appearance of the attractors. More such cases could have been included in this book but then its cost would have been considerably higher. You will probably henceforth want to view your three-dimensional attractors in color.

You will note that where one part of the attractor lies behind another you can see the more distant portion through the closer portion. Thus the attractor appears transparent which enables you to see its interior but tends to diminish the perception of depth. You might wish to modify the program so that the closer portion occludes the region behind it. It is relatively easy to do so using the BASIC POINT function to test the existing color of the pixel before plotting the new point and plotting it only if its color is higher in the sequence than the existing one. Thus each pixel will eventually be colored according to the closest part of the attractor. This effect can be accomplished by changing line 5160 of the program PROG14 to

5160 IF TRD% = 3 THEN C% = INT(NC% * (Z - ZMIN) / (ZMAX - ZMIN) + NC%) MOD NC%: IF POINT(XP YP) < C% THEN PSET (XP YP) C%

You can also alter the sequence of colors by changing the values stored in the array COLR% in line 5630. For example a sequence that mimics the rainbow would advance from red (12) through yellow (14) green (10) cyan (11) and blue (9) to magenta (13). With aerial perspective brilliant warm colors such as red appear closer to the viewer than lighter less brilliant cool colors such as blue which we associate with the distant sky. Thus assigning red to the large Z-values and blue to the small Z-values enhances the illusion of depth.

Most dialects of BASIC include a PALETTE command that allows you to change the screen colors without replotting the data but this command works differently with different versions of BASIC and in different graphics modes and so we will not try to provide a program that takes advantage of it. However a challenging programming exercise is to add the capability of rotating the color palette by pressing a key while a color attractor is being displayed to produce a psychedelic animated display. You might use the + key to rotate in one direction and the - key to rotate in the opposite direction. For most of the figures in this book the PALETTE command was used to interchange black and white before printing them to save ink and to improve the appearance of the attractors when they are displayed on a white background.

You may also wish to experiment with combining the various display techniques. Clearly there is nothing to preclude displaying a color attractor with shadows and contour bands. Such combinations offer interesting possibilities that will be exploited for the four-dimensional cases in the next chapter.

4.5 Characters

Many computer monitors and especially printers lack the capability of displaying colors. However it is often possible to produce a similar effect using a gray scale. In some cases the various colors are mapped automatically into a shade of gray. Another technique that will work on almost any computer and that offers interesting display possibilities is to map the Z-values into different ASCII characters and print them as a block of text. Such text files are easily manipulated by word processors transported to different computers displayed on almost any monitor and printed with any printer on paper of various sizes.

Perhaps the simplest method is to map the Z-values into consecutive ASCII characters thereby producing a type of gray scale with bands whose darkness depends on the density of the character. A more reasonable approach is to order the characters so that the more dense ones correspond to larger values of Z. The ordering depends on the font typeface and size of the characters. For example 14-point Courier bold can be ordered as shown in Table 4.2. The table uses thirty-two characters which is about the maximum for this technique since many of the characters have the same density. The eye can distinguish about 500 levels of gray. This sequence is only one of many that are equally good. To see the gray scale you should view the table from at least six feet away. Squinting and removing your glasses if you wear them might also help.

Table 4.2 Gray Scale Produced by Ordering 14-point Courier Bold Characters

.: ;"=>!/+?icl7IjvJL64VOASUDXEBM

.: ;"=>!/+?icl7IjvJL64VOASUDXEBM

.: ;"=>!/+?icl7IjvJL64VOASUDXEBM

.: ;"=>!/+?icl7IjvJL64VOASUDXEBM

.: ;"=>!/+?icl7IjvJL64VOASUDXEBM

.: ;"=>!/+?icl7IjvJL64VOASUDXEBM

.: ;"=>!/+?icl7IjvJL64VOASUDXEBM

.: ;"=>!/+?icl7IjvJL64VOASUDXEBM

.: ;"=>!/+?icl7IjvJL64VOASUDXEBM

.: ;"=>!/+?icl7IjvJL64VOASUDXEBM

.: ;"=>!/+?icl7IjvJL64VOASUDXEBM

.: ;"=>!/+?icl7IjvJL64VOASUDXEBM

.: ;"=>!/+?icl7IjvJL64VOASUDXEBM

.: ;"=>!/+?icl7IjvJL64VOASUDXEBM

The character densities for this table were determined by writing each of the ASCII characters from 33 to 127 (see Table 2.1) to the screen of a Macintosh computer and then counting the illuminated pixels using the POINT command in BASIC. Bold characters are chosen for their increased density. Avoid lower-case characters where possible since they often don't extend the full height and thus leave a wide blank band between their top and the bottom of the row above. Be sure to use a monospaced font rather than a proportional one. For the default font on most IBM computers in VGA mode SCREEN 12 a better sequence is:

.- ;=+>i%lI?v7zuCjTFSVGAEUDHBWQ

We will not develop a computer program for implementing this technique because the resolution is too poor for useful display on a computer screen or page of a book. Furthermore the program would be dependent on the fonts available and the capabilities of the printer. The programming is not difficult and closely parallels the example in PROG14.

However Table 4.3 provides an indication of what is possible using 7-point Courier bold characters with 78 lines of 103 characters each. The character sequence ordered by density for this case is given in Table 4.2. This case is a three-dimensional quadratic map with a code of ILRRHAEYWNTPWFLHTCSLYLFAKQITQTW.

At such a low resolution much of the detail is lost. However if you have a printer or plotter capable of printing small fonts on a large piece of paper you will be able to recover the resolution and produce figures of considerable artistic quality. You can divide the text into many segments on separate pages and tape them together or print them on a paper roll or fanfold paper to make a very skinny attractor many feet long. With ordinary objects such as the text in this book extreme stretching in one dimension just produces stick-like figures. However strange attractors are fractals and they have detail on all scales which ensures that they look interesting however much they are stretched.

Table 4.3 Three-Dimensional Quadratic Map with Character Scale

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JJ +///++???? icccccccciccciiiiiiii

JJv ++++??ii? ccllllllllllllllccccccccii

JJv ??+??iiii lll777777777777lllllllllccccc

JJJj ????iicci 7777777IIIII777777777777llllllccc

JJv ii?iiccc IIIIIIIIIIIIIIIIIIIIIIII7777777lllllc

JJvj ciiiclll IIjjjjjjjjjjjjjjjjjjjjIIIIIIII777777lllll

JLvjj cccicclll jjjjjjjvvvvvvvvvvvjjjjjjjjjjjjjIIIIII77777lll

LLvjI llcccl777 jvvvvvvvvvvvvvvvvvvvvvvvvvvvvjjjjjjjjIIIIII7777l

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LLvvjjI77llll77I7 vJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJvvvvvvvvjjjjjjIIIII7

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LLvvjjjII7777Ijj JJLLLLLLLLLLL LLLLLLLLLLLL6LLLLLLLLLLLLLLLLLJJJJJJJJvvvvvjjjjI

JLLvvjjIII77IIjjj LLLLLLL 6666666666666LLLLLLLLLLJJJJJJJvvvvvjj

LLvvvjjIIIIIIjvj LLLLL 66666666666666LLLLLLLLJJJJJJvvvvj

LLJvvjjjIIIIjjvv LLLLL 66666666666666LLLLLLLJJJJJJvv

LLvvvjjjjIIjjvv LLL 664446666666666LLLLLLJJJJ

LLJvvvjjjjjjjvvv L66 4444444466666666LLLLLLJ

JLJvvvvjjjjjvvJJ 6666 444444444446666666LLL

LLJvvvvvjjjvvJJ 666 4VV444444444466666

JLJJvv vvvvvvJJJ 66 VVVVVVV44444444

LLJJvv vvvvvJJJ 66 VVVVVVVVVVV444

JLJJJv vvvvJJJJ OOOOOVVVVVVVV

LLJJJv vJJJJL OOOOOOOOOOOOOOOOVV

LJJJJ JJJJLL AAAAAAAAAAAAAOOOOOOOOOOV

JJJJJJ JJJLL OOOOOOOOO AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAOO

JJJJJ LLLL OOOOOOOOOOOOOAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

JJJJJ LLL OOOOOAAAAAAAAAAAAAAAAAAAAAASSASAASSSSSSSSAAAAAAA

JJJJJJ LLL AAAAAAAAAAAAASSSSSSSSSSSSSSSSSSSSSSSSSSSS

JJJJJJ LLL AAAAAASSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS

JJJJJJ L66 AASSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS

JJJJJJ 66 ASSSSSSSSSSSSSSSUUUUUUUUUUUUUUUUUUUU

JJJJJJ 66 SSSSSSSSSSUUUUUUUUUUUUUUUUUUUUUUUU

LJJJJLL 6 SSSSSSUUUUUUUUUUUUUUUUUUUUUUUUUUU

LJJJLLL SSUUUUUUUUUUUUUUUUUUUUDDDDDDDDDDD

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LLLLLLLL UUUUUUDDDDDDDDDDDDDDDDDDDDDDDD

LLLLLLLLL UUUDDDDDDDDDDDDDDDDDXXXXXXXXXX

LLLLLLLLL UDDDDDDDDDDDDDDXXXXXXXXXXXXXX

LLLLLLLLL DDDDDDDDDDDXXXXXXXXXXXXXXXXXX

LLLLLLLL66 SS DDDDDDDDXXXXXXXXXXXXXXXXXXXX

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LLL666666444444VV SSSSS XXXXXXXXXXXXEEEEEEEEEEEEEE

LLL666664446644 SSSSS XXXXXXXXXEEEEEEEEEEEEEEEEE

LLL666666 SSSUU XXXXXXEEEEEEEEEEEEEEEBBBB

LLL6666 SUUUUU XXXXEEEEEEEEEEEEEBBBBBBBB

LLL66 O SUUUUUUU XXEEEEEEEEEEEEBBBBBBBBBB

LLLLL OO UUUUUUUUD EEEEEEEEEEEBBBBBBBBBBBBB

LLL OOO UUUUUUUUDDD EEEEEEEEBBBBBBBBBBBBBBB

LLLL OOOO SUUUUUUUDDDDDDD EEEEEEBBBBBBBBBBBBBMMM

LLLLLLL VOOOO SUUUUUUUDDDDDDDDXX EEEEEBBBBBBBBBBBBMMMMM

LLLLL6666664444444VVVVVOOOO SUUUUUUUUDDDDDDDDXXXXXX EEEBBBBBBBBBBBMMMMMMM

LLLLL6666664444444VVVVVOOO SSUUUUUUUUDDDDDDDXXXXXXXXXEEE EEBBBBBBBBBBBMMMMMMMM

LLLL6666666444444VVVVVVOO SSUUUUUUUDDDDDDDDXXXXXXXXEEEEEEEEEBBBBBBBBBBMMMMMMMMM

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LL666666444444VVVVVVOOO SSSSUUUUUUUDDDDDDDXXXXXXXXEEEEEEEEBBBBBBBBBB

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444444VVVVVVOOOOO ASSSSSSUUUUUUUDDDDDDDXXXXXXXEEEE

4VVVVVVOOOOOO AAASSSSSSUUUUUUUDDDDDDDXXXXXXXX

VVVOOOOOOOAAAAAASSSSSSUUUUUUUDDDDDDDXXXXX

VOOOOOOAAAAAASSSSSSUUUUUUUDDDDDDDXXX

OOOAAAAAASSSSSSSUUUUUUDDDDDDDD

AAAAAASSSSSSSUUUUUUDDDDD

AAASSSSSSSUUUUUUDDD

SSSSSUUUUUU

4.6 Anaglyphs

An alternate approach for displaying objects in three dimensions is the binocular stereogram in which the parallax produced by separate images in each eye creates the illusion of depth. The idea dates back to Socrates in the fourth century BC and the earliest stereograms were produced in the mid-1800's by Sir Charles Wheatstone and Sir David Brewster. The inception of motion pictures in the early 1900's was accompanied by 3-D movies using overlapping red-green images that were viewed through red-green glasses to produce a black-and-white image in what is called the anaglyphic process. Anaglyphs have also been widely used in comic books. Color 3-D movies using cross-Polaroid glasses were briefly popular in the 1950's. They require a special screen to reflect the polarized light from the projectors without allowing it to depolarize.

The anaglyphic process offers distinct advantages in computer visualization. The hardware requirements are minimal (a color monitor and a pair of 50-cent glasses) the programming is surprisingly simple and the results can be impressive. The main drawback is that the images produced are usually monochromatic although a gray scale and some limited coloration are possible.

Our perception of depth arises from a number of psychological and physiological processes. Many of these processes are induced by visual cues that don't depend on binocular vision such as the relative size and motion of objects interposition illumination shadows and focal accommodation. Others require the parallax attendant to stereoscopic vision. When some of the usual visual cues are absent or contradictory a rivalry ensues that demands time and mental effort for our brains to resolve. It is remarkable that with just the single cue of binocular stereopsis most people can quickly perceive a vivid three-dimensional image.

Consider a point at a distance D from the midpoint of your eyes whose separation we take to be e (typically 6.5 cm) as shown in Figure 4-51 (a). Assume the point is a single illuminated pixel on the computer screen. Each eye must swivel inward through an angle q in order for the two images to fuse into a single point where q is the angle whose tangent is e / 2D. It is this muscular response of the eyes that provides the brain with the relevant depth information.

Line drawing with parts (a) and (b)

Figure 4-51. Line of Sight of Each Eye when Viewing Anaglyphs

Now suppose you are to perceive the point to be at a distance D - Z from your eyes (a distance Z in front of the computer screen) as shown in Figure 4-51 (b). We must then plot two points on the computer screen separated horizontally by a distance d. From the similarity of the two triangles we calculate

d = eZ / (D-Z) (Eq. 4C)

The formula works also for negative Z.

To achieve the proper linear perspective we should plot the closer points a little farther apart than the more distant points but with unfamiliar objects such as strange attractors there is little reason to do so. The approximation of Equation 4C causes some expansion of the image for negative Z (behind the screen) and compression of the image for positive Z (in front of the screen). This compression can be desirable to keep the image always in front of your face rather than to let it pass behind your head.

The length of most people's arms is almost exactly ten times the distance between their eyes. Therefore a value of D / e = 10 is appropriate for a computer screen viewed at arm's length. In practice the viewing distance is not very critical. The perceived depth of the image is enhanced by viewing from a greater distance but it usually takes longer for the brain to accommodate and so it is often best to view first from close up. It sometimes speeds the adjustment to move your head from side to side.

A computer display optimized for viewing at arm's length is very effective when projected on a large screen and viewed in an auditorium. Were this not the case three-dimensional movies could not be shown to theater audiences. In such a case the brain perceives a scaled version of the image at a closer distance. The same effect occurs when viewing 2-D movies. The characters on the screen are not perceived as giants a large distance from the viewer. Similarly the brain is able to compensate almost without limit to other distortions if the objects are familiar. A movie viewed from the right-most seat in the front row appears normal after a short period of adjustment.

It is important to maintain a somewhat limited depth and field of view. Leonardo da Vinci recommended that a painting be optimally viewed from a distance equal to three times its width. Most computer screens approximately satisfy this criterion when viewed at arm's length. An object as deep as it is wide will thus require that the two images be separated by up to about an inch requiring that the eyes toe-in by about 3 degrees.

The computational task therefore is to plot each point that makes up the attractor twice with a horizontal separation proportional to the distance the point is to appear in front of or behind the screen and to arrange that one set of points be visible only to the left eye and the other only to the right eye. In the anaglyphic process this is done by plotting one set of points in red and the other in the complementary cyan (blue-green) and viewing through appropriate color-filtered glasses. By convention the left eye should only respond to the red and the right eye only to the cyan.

Note that individuals who are color blind should experience no difficulty since it is unnecessary (and indeed undesirable) to perceive the individual colors; it is only necessary that the eyes be sensitive to them. Certain other eye defects particularly those resulting in ocular asymmetry are more problematic.

You can plot the points on either a black or a white background. With a black background the images fuse into white (additive process) and with a white background the images fuse into black (subtractive process). The sense of Z is reversed with the choice of background. With a black background the red is seen through the red filter on the left eye while for a white background red is seen through the cyan filter on the right eye. In practice the white background is usually more satisfactory but you may want to try it both ways to see which works best for you. Wherever a red and cyan point overlap they should be plotted as a single black point if the background is white or as a single white point if the background is black. The changes required to the program to produce such anaglyphs are shown in the listing PROG15.

PROG15 Changes Required in PROG14 to Produce Anaglyphs

1000 REM THREE-D MAP SEARCH (With Anaglyphic Display)

1120 TRD% = 4                   'Display third dimension as anaglyph
3220    ZA = (ZMAX + ZMIN) / 2
3240    IF TRD% = 4 THEN LINE (XL
YL)-(XH
YH)
WH%
BF
3760 IF Q$ = "R" THEN TRD% = (TRD% + 1) MOD 5: T% = 3: IF N > 999 THEN N = 999: GOSUB 5600
4510        IF TRD% = 4 THEN PRINT "anaglyph  "
5170 IF TRD% <> 4 THEN GOTO 5240
5180    XRT = XP + XZ * (Z - ZA): C% = POINT(XRT
YP)
5190    IF C% = WH% THEN PSET (XRT
YP)
RD%
5200    IF C% = CY% THEN PSET (XRT
YP)
BK%
5210    XLT = XP - XZ * (Z - ZA): C% = POINT(XLT
YP)
5220    IF C% = WH% THEN PSET (XLT
YP)
CY%
5230    IF C% = RD% THEN PSET (XLT
YP)
BK%
5640 WH% = 15: BK% = 8: RD% = 12: CY% = 11

The program PROG15 assumes EGA or VGA graphics and a color monitor. If you have CGA graphics you can obtain satisfactory results in SCREEN 1 by changing the colors in line 5640 to WH% = 3: BK% = 0: RD% = 2: CY% = 1.

Some sample anaglyphs are shown in Plates 9 through 16. Use the special glasses included with the book. If these glasses are missing you can probably find a suitable pair at a comic book store. If you have difficulty acclimating to the anaglyphs try viewing them from close-up and then back away once you see the effect. You may need some practice especially since the attractors you are viewing are unfamiliar objects and they lack other depth clues. You might also try reversing the glasses (red over right eye) which reverses in and out.

Because of the large variation of computer monitor colors and spectacle filters ghost images are common. Manipulation of the computer color palette is of limited use since the monitor ultimately constructs its colors from three distinct phosphors (red green and blue). The usual problem is inadequate rejection of the green by the red filter resulting in a red ghost image when viewed against a white background. Suppression of the green by using only red and blue on a magenta background eliminates this problem but yields poor contrast of the resulting image. In some cases the ghost images can be suppressed by viewing through multiple pairs of glasses. You may want to adjust the intensity of the red green and blue so that the images seen by each eye through the glasses have similar intensities.

4.7 Stereo Pairs | Stereo Pairs

Believe it or not with a bit of practice you can learn to view attractors in 3-D without special glasses. For this purpose we print the two images side-by-side in the same color instead of superimposed on one another in different colors as we did with the anaglyphs. This technique permits full color displays and we will exploit this capability in the next chapter. For the moment let's consider only monochrome images.

First we develop the computer program necessary to produce the images. The images should not be separated more than the distance between your eyes which for most people is about 6.5 cm. If the images are separated by a larger distance the eyes have to rotate outward beyond the normal parallel position which at best is uncomfortable and at worst impossible. Such images are described as being walleyed. However if we reduce the images to a sufficiently small size on the computer screen the resolution will be poor. Therefore we will plot the images as large as possible and rely on the printer to reduce the size for comfortable viewing. If you prefer to sacrifice the resolution and view the attractors directly on the screen the program is written to make it easy for you to do so. Alternately your monitor may have an adjustment that allows you to shrink the width of the image.

The listing PROG16 shows the changes required to produce such stereo pairs.

PROG16 Changes Required in PROG15 to Produce Stereo Pairs

1000 REM THREE-D MAP SEARCH (With Stereo Display)

1120 TRD% = 5                   'Display third dimension as stereogram
3250    IF TRD% = 5 THEN LINE (XA
YL)-(XA
YH)
3320 IF PJT% = 1 AND TRD% < 5 THEN CIRCLE (XA
YA)
.36 * (XH - XL)
3760 IF Q$ = "R" THEN TRD% = (TRD% + 1) MOD 6: T% = 3: IF N > 999 THEN N = 999: GOSUB 5600
4520        IF TRD% = 5 THEN PRINT "stereogram"
5240 IF TRD% <> 5 THEN GOTO 5280
5250    HSF = 2                 'Horizontal scale factor
5260    XRT = XA + (XP + XZ * (Z - ZA) - XL) / HSF: PSET (XRT
YP)
5270    XLT = XA + (XP - XZ * (Z - ZA) - XH) / HSF: PSET (XLT
YP)
5280 RETURN

If you want to shrink the image for direct viewing from the computer screen change the horizontal scale factor (HSF) in line 5250 from 2 to a larger value that separates the images by about 6.5 centimeters or less if you are having trouble adapting to them.

Sample stereo pairs produced by this technique are shown in Figures 4-52 through 4-67. You should hold them exactly horizontally directly in front of your face at a normal reading distance and gaze into the distance until you see three images. The one in the middle should appear in 3-D and the ones on each side which you must train yourself to ignore should be in 2-D.

IGVQYNBN

Figure 4-52. Stereo Pair of Three-Dimensional Quadratic Map

IHJJDTIC

Figure 4-53. Stereo Pair of Three-Dimensional Quadratic Map

IJIFTNNC

Figure 4-54. Stereo Pair of Three-Dimensional Quadratic Map

IJVSURQU

Figure 4-55. Stereo Pair of Three-Dimensional Quadratic Map

ILNQAAYR

Figure 4-56. Stereo Pair of Three-Dimensional Quadratic Map

ILSQGYJJ

Figure 4-57. Stereo Pair of Three-Dimensional Quadratic Map

ILXRNRBK

Figure 4-58. Stereo Pair of Three-Dimensional Quadratic Map

IQAIENYA

Figure 4-59. Stereo Pair of Three-Dimensional Quadratic Map

ISYINLLU

Figure 4-60. Stereo Pair of Three-Dimensional Quadratic Map

IUWECTGS

Figure 4-61. Stereo Pair of Three-Dimensional Quadratic Map

IYBHCNQQ

Figure 4-62. Stereo Pair of Three-Dimensional Quadratic Map

JJICKAFX

Figure 4-63. Stereo Pair of Three-Dimensional Cubic Map

JNRVAPNY

Figure 4-64. Stereo Pair of Three-Dimensional Cubic Map

KLGFLNWT

Figure 4-65. Stereo Pair of Three-Dimensional Quartic Map

KLLRVFAK

Figure 4-66. Stereo Pair of Three-Dimensional Quartic Map

LTASNEPH

Figure 4-67. Stereo Pair of Three-Dimensional Quintic Map

You may find it difficult to adjust to the images at first but with practice you should be able to see them almost instantly. Viewing them should be relaxing with a sensation resembling a blank stare. Effort is required to return to normal viewing much like returning to the words on this page after gazing into the distance.

It might help to close your eyes momentarily and then reopen them if you are having trouble adapting. You can also buy an inexpensive hand stereoscope containing prisms that separate and magnify the images so that you can view them from a closer distance. Such viewing forces the side images out of your field of view and eases the adjustment to the middle image.

In the 1950's the View-Master stereo viewer was very popular for home use and many stereoscopic photographs were produced. Their popularity has waned and the View-Master is no longer made but similar inexpensive models can still be found in toy stores often with images of cartoon characters. Stereo images are usually photographed with a dual camera whose separation can be increased beyond the normal eye separation to enhance the depth sensation in what is called hyperstereo.

Stereoscopic images are used extensively by geologists and cartographers to determine terrain elevation from aerial photographs. As an airplane or satellite travels across the Earth photographs are taken at two positions separated by a distance much greater than the distance between the eyes. When viewed through a stereoscope the Earth appears as a scaled model viewed from just a foot or so above and it is easy to discern the elevation changes.

The preceding technique is called free viewing. An alternate and more difficult technique called short-focus viewing can also be used to view the stereo pairs. Here the procedure is to place the figure at arm's length but to look at a point about halfway to the figure. It may help to hold your finger at the halfway point and to cross your eyes until you see a single image of your finger. You should then see the three images of the figure float up off the page in the plane of your finger. The middle image should be three-dimensional. It may be difficult to keep the image from wavering and returning to the plane of the page. Squinting sometimes helps.

An advantage of short-focus viewing is that the images can be separated by a much larger distance and so it will work for projection on a large screen in a classroom or auditorium. However note that the image is in-out reversed from what it is with free viewing. This reversal is called pseudostereo. With anaglyphs pseudostereo can be obtained by reversing the glasses. Pseudostereo images would usually be very undesirable and would lead to all kinds of visual contradictions but with our strange attractors where there are no other visual depth cues it makes little difference. It can even be an advantage to be able to view the objects in either of these ways. Can you guess what you will see if you turn the figures upside down? Think about it and then give it a try.

You will note that with free viewing the left image disappears when you close your right eye and the right image disappears when you close your left eye. With short-focus viewing the opposite occurs. However it is incorrect to assume that each eye sees only one of the images. Both eyes see both images and the images fuse into one when your eyes are aimed in the proper direction.

4.8 Slices

We will discuss one final way to view attractors resulting from three-dimensional maps. Low-dimensional attractors are like loosely wound balls of string whereas high-dimensional ones are more like loaves of bread filled with holes. Anaglyphs and stereo pairs are effective for cases of low dimension but as the dimension increases the attractor becomes too opaque and the illusion of depth is lost.

Carrying the loaf-of-bread analogy a bit farther you could imagine slicing the loaf into a large number of very thin slices. The result is to decrease the dimension of the object by one. For example an object with a fractal dimension of 2.5 would become an object of dimension of 1.5 in each slice. This is an example of what is called a Poincaré section (as in "cross-section").

Perhaps it's easier to consider a specific case. Suppose the attractor were a loosely wound ball of very thin string. The attractor would then be essentially one-dimensional. The string would cross the slices at a number of points. Thus the slices would contain dots wherever the string pierced them. A set of a finite number of dots is an object of zero dimension.

If the attractor were a loosely crumpled piece of paper its dimension would be close to two. If you were to cut a thin slice through the crumpled paper you would be left with a handful of worm-like paper strings which are one-dimensional objects. You should now set the book down get a piece of paper and a pair of scissors and try it for yourself. Be sure to make the slice as thin as possible.

With our maps which contain only a finite number of points we cannot make the slices too thin lest they contain so few points as to be invisible. Furthermore it is impractical to look at all the slices if they are very thin because there are too many of them. As always we have to compromise. We'll use sixteen slices and lay them out in a 4 x 4 array so that we can see them all at once. On the computer screen this method entails a serious sacrifice in resolution but it does illustrate the principle. You might want to experiment with using a larger number of slices but displaying only a fraction of them. For example try using 64 slices and display every fourth one.

The modifications that are required to make the program produce a sliced display are shown in the listing PROG17.

PROG17 Changes Required in PROG16 to Produce Slices

1000 REM THREE-D MAP SEARCH (With Sliced Display)

1120 TRD% = 6                   'Display third dimension as slices
3260    IF TRD% <> 6 THEN GOTO 3310
3270        FOR I% = 1 TO 3
3280            XP = XL + I% * (XH - XL) / 4: LINE (XP
YL)-(XP
YH)
3290            YP = YL + I% * (YH - YL) / 4: LINE (XL
YP)-(XH
YP)
3300        NEXT I%
3760 IF Q$ = "R" THEN TRD% = (TRD% + 1) MOD 7: T% = 3: IF N > 999 THEN N = 999: GOSUB 5600
4530        IF TRD% = 6 THEN PRINT "slices    "
5280 IF TRD% <> 6 THEN GOTO 5330
5290    DZ = (15 * (Z - ZMIN) / (ZMAX - ZMIN) + .5) / 16
5300    XP = (XP - XL + (INT(16 * DZ) MOD 4) * (XH - XL)) / 4 + XL
5310    YP = (YP - YL + (3 - INT(4 * DZ) MOD 4) * (YH - YL)) / 4 + YL
5320    PSET (XP
YP)
5330 RETURN

Figures 4-68 through 4-83 show some sample attractors displayed as a succession of slices. The succession is from left to right and top to bottom in the same way you read (in most Western languages at least). In these cases the attractor dimension is greater than two; otherwise the dimension of the slices would be too small to be interesting.

IHORHNHP

Figure 4-68. Slices of Three-Dimensional Quadratic Map

IKRTYCFP

Figure 4-69. Slices of Three-Dimensional Quadratic Map

IMROJCCR

Figure 4-70. Slices of Three-Dimensional Quadratic Map

IRQEDYQN

Figure 4-71. Slices of Three-Dimensional Quadratic Map

JJOPUPCM

Figure 4-72. Slices of Three-Dimensional Cubic Map

JKKELGSD

Figure 4-73. Slices of Three-Dimensional Cubic Map

JLHREQNC

Figure 4-74. Slices of Three-Dimensional Cubic Map

JMGRCVAU

Figure 4-75. Slices of Three-Dimensional Cubic Map

KJJUPXHP

Figure 4-76. Slices of Three-Dimensional Quartic Map

KPBMLWTC

Figure 4-77. Slices of Three-Dimensional Quartic Map

KQEDOFHX

Figure 4-78. Slices of Three-Dimensional Quartic Map

KVFLHQBH

Figure 4-79. Slices of Three-Dimensional Quartic Map

LLVXALUX

Figure 4-80. Slices of Three-Dimensional Quintic Map

LMLGOLTV

Figure 4-81. Slices of Three-Dimensional Quintic Map

LNKNTCMV

Figure 4-82. Slices of Three-Dimensional Quintic Map

LPSONLEW

Figure 4-83. Slices of Three-Dimensional Quintic Map

In this chapter we have described a number of techniques whereby three-dimensional information can be exhibited on a computer screen or printed page. However none of these displays is truly three-dimensional. You have seen the term "3-D" used loosely in advertisements for computer graphics often meaning little more than a perspective drawing or a view from an oblique angle. In a true three-dimensional display the viewer must be able to see behind an object by moving his or her head from side to side. A holographic display allows this but most so-called 3-D displays do not. Anaglyphs and stereo pairs are probably better described as stereoscopic displays. They merely provide the illusion of 3-D as do shadows. Techniques using bands colors and slices deserve even less to be called 3-D however useful they are for conveying information about the third dimension. You should be appropriately discerning when confronted with graphics claimed to be 3-D.

IMDBUAMM

Plate 1: Three-Dimensional Quadratic Map

INDIJZID

Plate 2: Three-Dimensional Quadratic Map Projected on a Sphere

IOQRYLZU

Plate 3: Three-Dimensional Quadratic Map

IPGSYUPE

Plate 4: Three-Dimensional Quadratic Map Projected onto a Sphere

IRMZGEBC

Plate 5: Three-Dimensional Quadratic Map

IWHVUQKF

Plate 6: Three-Dimensional Quadratic Map Projected onto a Sphere

JNBVPGQK

Plate 7: Three-Dimensional Cubic Map

JOKDAGSW

Plate 8: Three-Dimensional Cubic Map Projected onto a Sphere

IFHTXHLV

Plate 9: Three-Dimensional Anaglyph (Use Colored Glasses)

IMYTXUJF

Plate 10: Three-Dimensional Anaglyph (Use Colored Glasses)

INBSGKIR

Plate 11: Three-Dimensional Anaglyph (Use Colored Glasses)

IPWRRDUT

Plate 12: Three-Dimensional Anaglyph (Use Colored Glasses)

JPCMNJSW

Plate 13: Three-Dimensional Anaglyph (Use Colored Glasses)

KKSLVGUQ

Plate 14: Three-Dimensional Anaglyph (Use Colored Glasses)

KLRLPKPH

Plate 15: Three-Dimensional Anaglyph (Use Colored Glasses)

LLOBPCEN

Plate 16: Three-Dimensional Anaglyph (Use Colored Glasses)

MMNGTRPW

Plate 17: Four-Dimensional Quadratic Map with Colored Bands

MOEYLVDU

Plate 18: Four-Dimensional Quadratic Map with Anaglyph Bands

MLKSEEJI

Plate 19: Four-Dimensional Quadratic Map with Shadowed Colors

MHBFVLWI

Plate 20: Four-Dimensional Quadratic Map with Banded Colors

MMDBTQOD

Plate 21: Stereo Pair of Four-Dimensional Quadratic Map

MLUICIOG

Plate 22: Slices of Four-Dimensional Quadratic Map

SFDUKQTR

Plate 23: Three-Dimensional Quartic ODE

RMEBBMXJ

Plate 24: Three-Dimensional Anaglyph of ODE (Use Colored Glasses)

VJEVASEY

Plate 25: Four-Dimensional Quadratic ODE with Colored Bands

WKLAAVLO

Plate 26: Four-Dimensional Quadratic ODE with Anaglyph Bands

VFQGCVAR

Plate 27: Four-Dimensional Quadratic ODE with Shadowed Colors

UOAUGGTF

Plate 28: Four-Dimensional Quadratic ODE with Banded Colors

UUHXPWNC

Plate 29: Stereo Pair of Four-Dimensional Quadratic ODE

XRDYAEOY

Plate 30: Slices of Four-Dimensional Quadratic ODE

]BDRSYI

Plate 31: Stochastic Web Map

]RWTKEE

Plate 32: Stochastic Web Map

CHAPTER 5

The Fourth Dimension

Although we normally think of space as three-dimensional mathematics is not so constrained. Strange attractors can be embedded in space of four and even higher dimensions. Their calculation is a straightforward extension of what we have done before. The challenge is to find ways to visualize such high-dimensional objects. This chapter will exploit a number of appropriate visualization techniques after a digression to explain why dimensions higher than three are useful for describing the world in which we live.

5.1 Hyperspace

Ordinary space is three-dimensional. The position of any point relative to an arbitrary origin can be characterized by a set of three numbers--the distance forward or back right or left up or down. An object such as a solid ball in this space may itself be three-dimensional or perhaps like an eggshell of negligible thickness it may be two-dimensional. You can also imagine an infinitely fine thread which is one-dimensional or the period at the end of this sentence which is essentially zero-dimensional. Although we can easily visualize objects with dimension less than or equal to 3 it is hard to envision space of higher dimension.

Before discussing the fourth dimension it is useful to clarify and refine some familiar terms. Perhaps the best example of a one-dimensional object is a straight line. The line may stretch to infinity in both directions or it may have ends. A line remains one-dimensional even if it bends in which case we call it a curve.

When we say that a curve is one-dimensional we are referring to its topological dimension. By contrast the Euclidean dimension is the dimension of the space in which the curve is embedded. If the line is straight both dimensions are one but if it curves the Euclidean dimension must be higher than the topological dimension. Both dimensions are integers. One definition of a fractal is an object whose Hausdorff-Besicovitch (fractal) dimension exceeds its topological dimension. For example a coastline on a flat map has a topological dimension of one a Euclidean dimension of two and a fractal dimension between one and two. It is an infinitely long line.

A special and important example of a curve is a circle which is curve of finite length but without ends every segment of which lies at a constant distance from a point at the center. Every circle lies in a plane which is a flat two-dimensional entity. Like a line the plane may stretch to infinity in all directions or it may have edges. If a plane has an edge we call it a disk. Note the distinction between a circle which is a one-dimensional object that does not include its interior and a disk which is a two-dimensional object that includes the interior.

Just as not all lines are straight not all two-dimensional objects are flat. A sheet of paper of negligible thickness remains two-dimensional if it is curled or even crumpled up in which case it is no longer a plane but it is still a surface. A curved surface has a Euclidean dimension of at least three. A surface can be finite but without edges. An example is a sphere every segment of which is at a constant distance from its center.

Note that just as a circle doesn't include its interior neither does a sphere. When we want to refer to the three-dimensional region bounded by a sphere we call it a ball. This terminology is universal among mathematicians but not among physicists who sometimes consider the dimension of circles and spheres to be the minimum dimension of the space in which they can be embedded (2 and 3 respectively).

Another example of a finite surface without edges is a torus most familiar as the surface of a doughnut or inner tube. Such curved spaces without edges are useful whenever one of the variables is periodic. Spaces of arbitrary dimensions whether flat or curved are called manifolds. The branch of mathematics that deals with these shapes is called topology.

If we could describe the world purely by specifying the position of objects three dimensions would suffice. However if you consider the motion of a baseball you are interested not only in where it is but in how fast it is moving and in what direction. Six numbers are needed to specify both its position and its velocity. This six-dimensional space is called phase space. Furthermore if the balls are spinning six more dimensions are needed for each ball one to specify the angle and another the angular velocity about each of three perpendicular axes through the ball.

If you have two balls that move independently you need a phase space with twice as many (twelve) dimensions and so forth. Contemplate the phase-space dimension required to specify the motion of more than 1025 molecules in every cubic meter of air! Sometimes physicists even find it useful to perform calculations in an infinite-dimensional space called Hilbert space.

You might also be interested in other properties of the balls such as their temperature color or radius. Thus the state of the balls as time advances can be described by a curve or trajectory in some high-dimensional space called state space in which the various perpendicular directions correspond to the quantities that describe the balls. The trajectory is a curve connecting temporally successive points in state space.

You have probably heard of time referred to as the fourth dimension and associate the idea with the theory of relativity. Long before Einstein it was obvious that to specify an event as opposed to a location it is necessary to specify not only where the event occurred (X Y and Z) but when (T). We can consider events to be points in this four-dimensional space.

Note that the spatial coordinates of a point are not unique. An object four feet in front of one observer might be six feet to the right of a second and two feet above a third. The values of X Y and Z of the position depend on where the coordinate system is located and how it is oriented. However we would expect the various observers to agree on the separation between any two locations. Similarly we expect all observers to agree on the time interval between two events.

The special theory of relativity asserts that observers usually do not agree on either the separation or the time interval between two events. Events that are simultaneous for one observer will not be simultaneous for a second moving relative to the first. Similarly two successive events at the same position as seen by one observer will be seen at different positions by the other.

You have probably heard that according to the special theory of relativity moving clocks run slow and moving meter sticks are shortened. (It is also true that the effective mass of an object increases when it moves leading to the famous E = mc2 but that's another story.) These discrepancies remain even after the observers correct for their motion and for the time required for the information about the events to reach them traveling at the speed of light. It is important to understand that these facts have nothing to do with the properties of clocks and meter sticks and that they are not illusions; they are properties of space and time neither of which possess the absolute qualities we normally ascribe to them.

What is remarkable is that all observers agree on the separation between the events in four-dimensional space-time. This separation is called the proper length and it is calculated from the Pythagorean theorem by taking the square root of the sum of the squares of the four components after converting the time interval (ÆT) to a distance by multiplying it by the speed of light (c). The only subtlety is that the square of the time enters as a negative quantity:

Proper length = [ÆX2 + ÆY2 + ÆZ2 - c2ÆT2]1/2 (Eq. 5A)

Because of the minus sign time is considered to be an imaginary dimension since an imaginary number is one whose square is negative. Note however that imaginary does not mean it is any less real than the other dimensions only that its square combines with the others through subtraction rather than through addition. If you are unfamiliar with imaginary numbers don't be put off by the name. They aren't really imaginary; they are just the other part of certain quantities that require a pair of numbers rather than a single number to specify them.

The minus sign also means that proper length unlike ordinary length may be imaginary. If the proper length is imaginary we say the events are separated in a time-like as opposed to a space-like manner. Time-like events can be causally related (one event can influence the other) but space-like events cannot because information about one would have to travel faster than the speed of light to reach the other which is impossible. Events separated in a time-like manner are more conveniently characterized by a proper time:

Proper time = [ÆT2 - ÆX2/c2 - ÆY2/c2 - ÆZ2/c2]1/2 (Eq. 5B)

In this case time is real but space is imaginary. Proper length is the length of an object as measured by an observer moving with the same velocity as the object and proper time is the time interval as measured by a clock moving with the same velocity as the observer.

Quantities such as proper length and proper time on which all observers agree independent of their motion are called invariants. The speed of light itself is an invariant. There are many others and they all involve four components that combine by the Pythagorean theorem.

Thus the theory of relativity ties space and time together in a very fundamental way. One person's space is another person's time. Since space and time can be traded back and forth there is no reason to call time the fourth dimension any more than we call width the second dimension. It is better just to say that space-time is four-dimensional with each dimension on an equal footing. The apparent asymmetry between space and time comes from the large value of c (3 x 108 meters per second or about a billion miles per hour) and the fact that time moves in only one direction (past to future). It is also important to understand that although special relativity is called a theory it has been extensively verified to high accuracy by many experiments most of which involve particle accelerators.

The foregoing discussion motivates why it might be useful to consider four-dimensional space and four-dimensional objects but it is probably fruitless to waste too much time trying to visualize them. However we can describe them mathematically as extensions of familiar objects in lower dimensions.

For example a hypercube is the four-dimensional extension of the three-dimensional cube and the two-dimensional square. It has 16 corners 32 edges 24 faces and contains 8 cubes. Its hypervolume is the fourth power of the length of each edge just as the volume of a cube is the cube of the length of an edge and the area of a square is the square of the length of an edge.

A hypersphere consists of all points at a given distance from its center in four-dimensional space. Its hypersurface is three-dimensional and consists of an infinite family of spheres just as the surface of an ordinary sphere is two-dimensional and consists of an infinite family of circles. There is reason to believe that our Universe might be a hypersurface of a very large hypersphere in which case we could see ourselves if we peered far enough into space except for the fact that we are also looking backwards to a time before the Earth existed. We would also need an incredibly powerful telescope to see the Earth in this way. Thus our perception that space is three-dimensional could be analogous to the ancient view that the Earth was flat a consequence of experience limited to a small portion of its surface.

5.2 Projections

The previous section was intended to motivate the consideration of strange attractors embedded in four-dimensional space but most of the discussion there is not essential to what follows. We will now describe the computer program necessary to produce attractors in four dimensions and then develop methods to visualize them.

The mathematical generalization from three to four dimensions is straightforward. Whereas before we had three variables--X Y and Z--we now have a fourth. Having used up the three letters at the end of the alphabet we will back up and use W for the fourth dimension but remember that all the dimensions are on an equal footing. We will use the first letters M N O and P to code 4-D attractors of second through fifth order respectively. The number of coefficients for these cases is 60 140 280 and 504 respectively. The number of coefficients for order O is (O + 1)(O + 2)(O + 3)(O + 4) / 6. The number of four-dimensional fifth-order codes is 25504 a number too large to compare to anything meaningful; it might as well be infinite.

The program modifications required to add a fourth dimension are shown in the listing PROG18.

PROG18 Changes Required in PROG17 to add a Fourth Dimension

1000 REM FOUR-D MAP SEARCH

1020 DIM XS(499)
YS(499)
ZS(499)
WS(499)
A(504)
V(99)
XY(4)
XN(4)
COLR%(15)
1070 D% = 4                     'Dimension of system
1120 TRD% = 0                   'Display third dimension as projection
1540 W = .05
1550 XE = X + .000001: YE = Y: ZE = Z: WE = W
1610 WMIN = XMIN: WMAX = XMAX
1720 M% = 1: XY(1) = X: XY(2) = Y: XY(3) = Z: XY(4) = W
2010 M% = M% - 1: XNEW = XN(1): YNEW = XN(2): ZNEW = XN(3): WNEW = XN(4)
2180    IF W < WMIN THEN WMIN = W
2190    IF W > WMAX THEN WMAX = W
2210 XS(P%) = X: YS(P%) = Y: ZS(P%) = Z: WS(P%) = W
2410 IF ABS(XNEW) + ABS(YNEW) + ABS(ZNEW) + ABS(WNEW) > 1000000! THEN T% = 2
2470 IF ABS(XNEW - X) + ABS(YNEW - Y) + ABS(ZNEW - Z) + ABS(WNEW - W) < .000001 THEN T% = 2
2540 W = WNEW
2910 XSAVE = XNEW: YSAVE = YNEW: ZSAVE = ZNEW: WSAVE = WNEW
2920 X = XE: Y = YE: Z = ZE: W = WE: N = N - 1
2950 DLZ = ZNEW - ZSAVE: DLW = WNEW - WSAVE
2960 DL2 = DLX * DLX + DLY * DLY + DLZ * DLZ + DLW * DLW
3010    ZE = ZSAVE + RS * (ZNEW - ZSAVE): WE = WSAVE + RS * (WNEW - WSAVE)
3020    XNEW = XSAVE: YNEW = YSAVE: ZNEW = ZSAVE: WNEW = WSAVE
3150 IF WMAX - WMIN < .000001 THEN WMIN = WMIN - .0000005: WMAX = WMAX + .0000005
3680 IF Q$ = "D" THEN D% = 1 + (D% MOD 4): T% = 1
3920 IF N = 1000 THEN D2MAX = (XMAX - XMIN) ^ 2 + (YMAX - YMIN) ^ 2 + (ZMAX - ZMIN) ^ 2 + (WMAX - WMIN) ^ 2
3940 DX = XNEW - XS(J%): DY = YNEW - YS(J%): DZ = ZNEW - ZS(J%): DW = WNEW - WS(J%)
3950 D2 = DX * DX + DY * DY + DZ * DZ + DW * DW
4760 IF D% > 2 THEN FOR I% = 3 TO D%: M% = M% / (I% - 1): NEXT I%

If you run the program PROG18 under certain old versions of BASIC such as BASICA and GW-BASIC you are likely to get an error in line 2710 when the program attempts to construct a code for the fourth-order and fifth-order maps as a result of the string-length limit of 255 characters. In such a case you may need to restrict the search to second and third orders by setting OMAX% = 3 in line 1060. Alternately it's not difficult to modify the program to store the code in a pair of strings or to replace the string with a one-dimensional array of integers containing the numeric equivalents of each character in the string perhaps with a terminating zero to signify the end of the string. Thus for example line 2710 would become

2710 CODE%(I%) = 65 + INT(25 * RAN)

and line 2740 would become

2740 A(I%) = (CODE%(I%) - 77) / 10

after dimensioning CODE%(504) in line 1020.

You will also notice that the search for attractors is painfully slow unless you have a very fast computer and a good compiler. You should refer to Table 2.2 where some options for increasing the speed are examined. The search can be made faster by limiting it to second order by setting OMAX% = 2 in line 1060.

There is another trick to increase drastically the rate at which four-dimensional strange attractors are found without sacrificing variety. It turns out that most of these attractors have their constant terms near zero. The reason presumably has to do with the fact that the origin (X = Y = Z = W = 0) is then a fixed point and the initial condition is chosen near the origin (X0 = Y0 = Z0 = W0 = 0.05). If the fixed point is unstable then we have one of the conditions necessary for chaos. It is easy to accomplish this by adding after line 2730 a statement such as:

2735 IF I% MOD M% / D% = 1 THEN MID$(CODE$ I% + 1 1) = "M"

The result is to increase the rate of finding attractors by about a factor of fifty. Many of the attractors illustrated in this chapter were produced in this way. This change will also increase the rate for lower-dimensional maps but by a much smaller factor. This improvement suggests that there is yet room to optimize the search routine by a more intelligent choice of the values of the other coefficients.

Note that the program PROG18 does not attempt to display the fourth dimension but rather projects it onto the other three for which all the visualization techniques of the last chapter are available. Don't waste too much time trying to understand what it means to project a four-dimensional object onto a three-dimensional space. It is just a generalization of projecting a three-dimensional object onto a two-dimensional surface. In the program it simply involves plotting X Y and Z and ignoring the variable W.

Some examples of four-dimensional attractors projected onto the two-dimensional XY-plane are shown in Figures 5-1 through 5-20. They don't look particularly different from those obtained by projecting three-dimensional attractors onto the plane or indeed by just plotting two-dimensional attractors directly. Note that most of these attractors have fractal dimensions less than or about 2.0 and so perhaps it is not too surprising that their projections resemble those produced by equations of lower dimension. It is rare to find attractors with fractal dimensions greater than 3.0 produced by four-dimensional polynomial maps as will be shown in Section 8.1.

MIJATEND

Figure 5-1. Projection of Four-Dimensional Quadratic Map

MJRMIKPD

Figure 5-2. Projection of Four-Dimensional Quadratic Map

MLAEODJG

Figure 5-3. Projection of Four-Dimensional Quadratic Map

MLDFKCIK

Figure 5-4. Projection of Four-Dimensional Quadratic Map

MLOGSXIH

Figure 5-5. Projection of Four-Dimensional Quadratic Map

MLWLRTTL

Figure 5-6. Projection of Four-Dimensional Quadratic Map

MMCRYBHO

Figure 5-7. Projection of Four-Dimensional Quadratic Map

MMRPEQPU

Figure 5-8. Projection of Four-Dimensional Quadratic Map

MMSHIOLQ

Figure 5-9. Projection of Four-Dimensional Quadratic Map

MMTRGGKK

Figure 5-10. Projection of Four-Dimensional Quadratic Map

MMUCGTWN

Figure 5-11. Projection of Four-Dimensional Quadratic Map

MMVMLIKD

Figure 5-12. Projection of Four-Dimensional Quadratic Map

MMWDKKHN

Figure 5-13. Projection of Four-Dimensional Quadratic Map

MMWIJHYU

Figure 5-14. Projection of Four-Dimensional Quadratic Map

MMXBCTVQ

Figure 5-15. Projection of Four-Dimensional Quadratic Map

MNGPYOCM

Figure 5-16. Projection of Four-Dimensional Quadratic Map

MPGPMWGT

Figure 5-17. Projection of Four-Dimensional Quadratic Map

NMHHSPLA

Figure 5-18. Projection of Four-Dimensional Cubic Map

NMMQXNNY

Figure 5-19. Projection of Four-Dimensional Cubic Map

OKPMLHFL

Figure 5-20. Projection of Four-Dimensional Quartic Map

5.3 Other Display Techniques

Projecting two of the four dimensions onto the remaining two is akin to buying a Ferrari to make trips to the grocery store. Much of our effort is wasted. We need to use the techniques developed in the last chapter to display three dimensions and devise additional methods to display simultaneously the fourth dimension.

Since we have several methods for displaying three dimensions we should be able to use some of them in combination to visualize all four dimensions. Table 5.1 summarizes the display techniques that we have used and indicates the number of dimensions that can be visualized with various combinations of them. In the table a dash indicates that the combination is not possible and a question mark indicates that the combination is possible but leads to contradictory visual information.

Table 5.1 Combinations of Display Techniques and the Number of Dimensions that can be Visualized with Each

Third Dimension

Project Shadow Bands Color Anaglyph Stereo Slices

4th Dim

Project 2D 3D 3D 3D 3D 3D 3D

Shadow 3D - 4D 4D ? ? 4D

Bands 3D 4D ? 4D 4D 4D 4D

Color 3D 4D 4D - - 4D 4D

Anaglyph 3D ? 4D - - ? 4D

Stereo 3D ? 4D 4D ? - 4D

Slices 3D 4D 4D 4D 4D 4D -

In Table 5.1 certain of the entries are in boldface. These are the ones we will implement in the program. They were chosen because of their visual effectiveness ease of programming and lack of redundancy with other combinations. Cases below and to the left of the diagonal duplicate those above and to the right. The changes needed in the program to produce such four-dimensional displays are shown in the listing PROG19.

PROG19 Changes Required in PROG18 to Display the Fourth Dimension

1000 REM FOUR-D MAP SEARCH (With 4-D Display Modes)

1040 PREV% = 5                  'Plot versus fifth previous iterate
1120 TRD% = 1                   'Display third dimension as shadow
1130 FTH% = 2                   'Display fourth dimension as colors
3630 IF Q$ = "" OR INSTR("ADHIPRSX"
Q$) = 0 THEN GOSUB 4200
3720 IF Q$ = "H" THEN FTH% = (FTH% + 1) MOD 3: T% = 3: IF N > 999 THEN N = 999: GOSUB 5600
4330    PRINT TAB(27); "H: Fourth dimension is ";
4340        IF FTH% = 0 THEN PRINT "projection"
4350        IF FTH% = 1 THEN PRINT "bands     "
4360        IF FTH% = 2 THEN PRINT "colors    "
5010 C4% = WH%
5020 IF D% < 4 THEN GOTO 5050
5030    IF FTH% = 1 THEN IF INT(30 * (W - WMIN) / (WMAX - WMIN)) MOD 2 THEN GOTO 5330
5040    IF FTH% = 2 THEN C4% = 1 + INT(NC% * (W - WMIN) / (WMAX - WMIN) + NC%) MOD NC%
5050 IF D% < 3 THEN PSET (XP
YP): GOTO 5330    'Skip 3-D stuff
5060 IF TRD% = 0 THEN PSET (XP
YP)
C4%
5080    IF D% > 3 AND FTH% = 2 THEN PSET (XP
YP)
C4%: GOTO 5110
5130 IF TRD% <> 2 THEN GOTO 5160
5140    IF D% > 3 AND FTH% = 2 AND (INT(15 * (Z - ZMIN) / (ZMAX - ZMIN) + 2) MOD 2) = 1 THEN PSET (XP
YP)
C4%
5150    IF D% < 4 OR FTH% <> 2 THEN C% = COLR%(INT(60 * (Z - ZMIN) / (ZMAX - ZMIN) + 4) MOD 4): PSET (XP
YP)
C%
5260    XRT = XA + (XP + XZ * (Z - ZA) - XL) / HSF: PSET (XRT
YP)
C4%
5270    XLT = XA + (XP - XZ * (Z - ZA) - XH) / HSF: PSET (XLT
YP)
C4%
5320    PSET (XP
YP)
C4%
5630 IF TRD% = 3 OR (D% > 3 AND FTH% = 2 AND TRD% <> 1) THEN FOR I% = 0 TO NC%: COLR%(I%) = I% + 1: NEXT I%

In presenting sample displays from PROG19 we ignore those that convey only three-dimensional information and concentrate on the new combinations that permit full four-dimensional displays. They fall into two groups--those that require the use of color and those that do not. Examples of the three 4-D monochrome combinations are shown in Figures 5-21 through 5-44 and examples of the six color combinations are shown in Plates 17 through 22.

MGDGPSEL

Figure 5-21. Four-Dimensional Quadratic Map with Shadow Bands

MMBDLBAK

Figure 5-22. Four-Dimensional Quadratic Map with Shadow Bands

MMDWDKNS

Figure 5-23. Four-Dimensional Quadratic Map with Shadow Bands

MMUDTNUF

Figure 5-24. Four-Dimensional Quadratic Map with Shadow Bands

MMWHELTG

Figure 5-25. Four-Dimensional Quadratic Map with Shadow Bands

MOFUUMXC

Figure 5-26. Four-Dimensional Quadratic Map with Shadow Bands

MUACOEKV

Figure 5-27. Four-Dimensional Quadratic Map with Shadow Bands

NMTICQDD

Figure 5-28. Four-Dimensional Quadratic Map with Shadow Bands

MKAIUOAY

Figure 5-29. Four-Dimensional Quadratic Map with Stereo Bands

MKWWFTQH

Figure 5-30. Four-Dimensional Quadratic Map with Stereo Bands

MNGFOFMK

Figure 5-31. Four-Dimensional Quadratic Map with Stereo Bands

NMCSOXFI

Figure 5-32. Four-Dimensional Cubic Map with Stereo Bands

NMCYETOQ

Figure 5-33. Four-Dimensional Cubic Map with Stereo Bands

NMGQIXCG

Figure 5-34. Four-Dimensional Cubic Map with Stereo Bands

OMOQRUJS

Figure 5-35. Four-Dimensional Quartic Map with Stereo Bands

OMXLIBUD

Figure 5-36. Four-Dimensional Quartic Map with Stereo Bands

MMICFXPC

Figure 5-37. Four-Dimensional Quadratic Map with Sliced Bands

MMKCVPGE

Figure 5-38. Four-Dimensional Quadratic Map with Sliced Bands

MMPIVFOB

Figure 5-39. Four-Dimensional Quadratic Map with Sliced Bands

MUDPYXTB

Figure 5-40. Four-Dimensional Quadratic Map with Sliced Bands

NMHYYGLN

Figure 5-41. Four-Dimensional Cubic Map with Sliced Bands

OMMOXNKS

Figure 5-42. Four-Dimensional Quartic Map with Sliced Bands

OMPRETYL

Figure 5-43. Four-Dimensional Quartic Map with Sliced Bands

PMRPHCDN

Figure 5-44. Four-Dimensional Quintic Map with Sliced Bands

You might be interested in the challenge of producing attractors embedded in dimensions higher than four. In five dimensions you need to define a new variable say V and modify the program as was done for four dimensions in PROG18. The program has been written to make it relatively easy to extend it to five or even higher dimensions. Be forewarned that the calculation will be very slow. You will almost certainly want to set the coefficients of the constant terms to zero and probably restrict your search to quadratic maps. The number of fifth-dimension polynomial coefficients for order O is (O + 1)(O + 2)(O + 3)(O + 4)(O + 5) / 24. With O = 5 the number is 1260.

The simplest display technique is to project the fifth dimension onto the other four. This is what the program does automatically if you don't do anything special. There are several combinations of techniques which we have already developed that are capable of displaying five dimensions. You might try combining shadows bands and color for example. Table 5.2 lists the seven possible combinations of five-dimensional display techniques that don't lead to visual contradictions.

Table 5.2 Combinations of Display Techniques That Can Be Used in Five Dimensions

For a heroic exercise in programming visualization and patience you can try to extend the calculation to six dimensions. A six-dimensional fifth-order system of polynomials has 2772 coefficients. There are only two appropriate combinations of display techniques suitable for six dimensions: shadow-bands-color-slices and bands-color-stereo-slices. If you decide to try seven dimensions you will have to invent a new display technique.

5.4 Writing on the Wall

Since four-dimensional attractors have the greatest complexity and variety of all the cases described in this book they offer the greatest potential as display art For such purposes you will probably want to print them on a large sheet of paper. With an appropriate printer or plotter any of the visualization techniques previously described can be used to produce such large prints.

An alternate technique that has proved very successful is an extension of the character-based method described in Section 4.5. In this technique the third dimension is coded as an ASCII character with a density related to the Z-value and the fourth dimension is coded in color. Color pen and pencil plotters and ink-jet plotters as well as more expensive but high quality electrostatic and thermal plotters normally used for engineering and architectural drawings can print text on sheets up to 36 inches wide. Ink-jet plotters are growing in popularity over the more traditional pen plotters because they are faster and quieter and don't require special paper. They can also print gray scales. With care you can piece together smaller segments printed by more conventional means.

When the attractors are reduced to sequences of text resolutions of 640 by 480 (VGA) or 800 by 600 (Super VGA) produce large figures whose individual characters can be read when examined closely but that blend into continuous contours when viewed from a distance. Artists often use this technique in which the viewer is provided with a different visual experience on different scales. You should use the largest and boldest characters available to maximize the contrast provided they remain readable. There should be little or no space between rows and columns of characters. With a pen plotter the pen size can be chosen for the best compromise of contrast and readability. A pen that makes a line width of 0.35 mm (fine) is a reasonable choice.

Inks are available in only a limited number of colors and pen plotters are usually capable of accommodating only a small number of pens. The pens can be sequenced to place compatible colors next to one another. With eight pens and commonly available inks a good sequence is magenta red orange (or yellow) brown black green turquoise and blue. The closest color sequence for viewing on the computer screen from Table 4.1 is 13 12 4 (or 14) 6 8 2 3 and 9 with a white (15) background. With upwards of twenty characters producing different color intensities the limitation of eight colors of ink is not a serious one. With eight colors and ASCII codes from 32 to 255 you can have 28 different intensities for each color. The inks can be mixed to produce different shades of the colors. Pencils are less expensive and don't clog or dry out as pens often do but pencil plots have a tendency to smudge. Ink of course will also smudge until it is thoroughly dry. Plotters are relatively slow and attractors produced by this method typically require a few hours to a full day to produce.

Paper commonly used for engineering drawings comes in at least five standard sizes--A (8 1/2 by 11 inch) B (12 by 18 inch) C (18 by 24 inch) D (24 by 36 inch) and E (36 by 48 inch). English sizes and architectural sizes are slightly different and thus a sheet may vary somewhat from these dimensions. Thirty-six-inch-wide paper is also available on long rolls.

Common paper types are tracing bond which is the most economical vellum which is smooth and translucent and polyester film which is highly translucent dimensionally stable and relatively expensive. The translucent papers offer the interesting possibility of backing the print with a monochrome or color copy of itself to enhance the contrast or to produce a shadow effect if the two are displaced slightly. Other interesting effects can be achieved by backing one translucent attractor with a print of another or by back-lighting the print. Some of the papers stretch slightly and thus have a tendency to wrinkle. Paper with significant acid content should be avoided since it turns yellow and becomes brittle with age.

Some of the most artistic examples of strange attractors have been produced by these techniques but they cannot be adequately illustrated in this book. No computer program is offered since it is so dependent on your hardware. You will want to experiment to find the technique that works best for you and that makes the most effective use of your printer or plotter.

5.5 Murals and Movies

The technique of making large-scale attractors for display can be carried to its logical extreme by making a mural. Special techniques using some type of stencil are required to transform the computer output to paint on the wall. Silk screen is useful for transferring the image to fabrics. Fractal tee-shirts employing this technique have recently become popular.

To produce a mural you will probably need to start with a large number of plots each showing a small section of the attractor. A property of fractals is that they have detail on all scales and thus a large mural should look interesting when viewed either from a distance or close up.

You might also photograph the computer screen or a high quality print and produce slides that can be projected onto a large surface or screen with a slide projector. Equipment is available commercially for producing slides directly from digital computer output. A sequence of such slides makes a very compelling presentation or visual accompaniment to a lecture or musical production.

The color slices as shown in Plate 22 suggest the possibility of making color movies by extending the technique to a very large number of slices and using each one as a frame of a movie. The effect is to cause the attractor to emerge at a point in an empty field and to grow slowly bending and wiggling until fully developed and then to disappear slowly into a different point. If the technology for doing this is unavailable you might try printing a large number of attractor slices on small cards and fanning through them to produce a semblance of animation. This technique with attractors described in Section 7.6 was used to produce the animation in the upper-right corner of the odd pages of this book.

If the idea of making strange-attractor movies appeals to you another technique is to take one of your favorite attractors and slowly change one or more of the coefficients in successive frames of the movie. A good way to start is to multiply all the coefficients by a factor that varies from slightly less than 1.0 to slightly greater than 1.0. You will need to determine the range over which the coefficients can be changed without the solutions becoming unbounded or non-chaotic. The ends of this range then become the beginning and end of the movie.

Sometimes the attractor slowly and continuously alters its shape. The changes can involve bifurcations such as the period-doubling sequence in the logistic equation described in Chapter 1. Such bifurcations are called subtle. Other times the attractor and its basin will abruptly disappear at a critical value of the control parameter. Such discontinuous bifurcations are called catastrophes.

If the control parameter is changed in the opposite direction the result may be different from simply running the movie backwards. This is an example of hysteresis which is a form of memory in a dynamical system. It serves to limit the occurrence of catastrophes. The thermostat that controls your heat probably uses hysteresis to keep the furnace from cycling on and off too frequently. Catastrophic bifurcations usually exhibit hysteresis whereas subtle bifurcations do not.

These four-dimensional maps are also well suited for color holographic display or for experimentation with virtual reality in which the view is controlled by motion of one's head and hands to give the sensation of moving through the object. The technology is complicated but the results should be visually and mentally stimulating.

5.6 Search and Destroy

If you have worked carefully through the text your program will have created a disk file SA.DIC containing the codes of all the attractors generated since you ran the program PROG11. We will now develop the capability to examine these attractors and save the interesting ones in a file FAVORITE.DIC while discarding the others. This feature will allow you to run the program overnight and collect attractors for rapid viewing the next day. This capability is especially useful if you have a slow computer. The required program changes are shown in the listing PROG20.

PROG20 Changes Required in PROG19 to Evaluate the Attractors in SA.DIC and Save the Best of Them in FAVORITE.DIC

1000 REM FOUR-D MAP SEARCH (With Search and Destroy)

1380 IF QM% <> 2 THEN GOTO 1420
1390    NE = 0: CLOSE
1400    OPEN "SA.DIC" FOR APPEND AS #1: CLOSE
1410    OPEN "SA.DIC" FOR INPUT AS #1
2420 IF QM% = 2 THEN GOTO 2490  'Speed up evaluation mode
2610 IF QM% <> 2 THEN GOTO 2640 'Not in evaluate mode
2620    IF EOF(1) THEN QM% = 0: GOSUB 6000: GOTO 2640
2630    IF EOF(1) = 0 THEN LINE INPUT #1
CODE$: GOSUB 4700: GOSUB 5600
3340 IF QM% <> 2 THEN GOTO 3400         'Not in evaluate mode
3350    LOCATE 1
1: PRINT "<Space Bar>: Discard       <Enter>: Save";
3370    LOCATE 1
49: PRINT "<Esc>: Exit";
3380    LOCATE 1
69: PRINT CINT((LOF(1) - 128 * LOC(1)) / 1024); "K left";
3390    GOTO 3430
3620 IF QM% = 2 THEN GOSUB 5800                 'Process evaluation command
3630 IF INSTR("ADEHIPRSX"
Q$) = 0 THEN GOSUB 4200
3710 IF Q$ = "E" THEN T% = 1: QM% = 2
4220 WHILE Q$ = "" OR INSTR("AEIX"
Q$) = 0
4320    PRINT TAB(27); "E: Evaluate attractors"
5800 REM Process evaluation command
5810 IF Q$ = " " THEN T% = 2: NE = NE + 1: CLS
5820 IF Q$ = CHR$(13) THEN T% = 2: NE = NE + 1: CLS : GOSUB 5900
5830 IF Q$ = CHR$(27) THEN CLS : GOSUB 6000: Q$ = " ": QM% = 0: GOTO 5850
5840 IF Q$ <> CHR$(27) AND INSTR("HPRS"
Q$) = 0 THEN Q$ = ""
5850 RETURN
5900 REM Save favorite attractors to disk file FAVORITE.DIC
5910 OPEN "FAVORITE.DIC" FOR APPEND AS #2
5920 PRINT #2
CODE$
5930 CLOSE #2
5940 RETURN
6000 REM Update SA.DIC file
6010 LOCATE 11
9: PRINT "Evaluation complete"
6020 LOCATE 12
8: PRINT NE; "cases evaluated"
6030 OPEN "SATEMP.DIC" FOR OUTPUT AS #2
6040 IF QM% = 2 THEN PRINT #2
CODE$
6050 WHILE NOT EOF(1): LINE INPUT #1
CODE$: PRINT #2
CODE$: WEND
6060 CLOSE
6070 KILL "SA.DIC"
6080 NAME "SATEMP.DIC" AS "SA.DIC"
6090 RETURN

The program uses the E key to enter the evaluation mode. When in this mode the attractors in SA.DIC are displayed one by one. Each case remains on the screen and continues to iterate until you press the Space Bar which deletes it the Enter key which saves it in the file FAVORITE.DIC the Esc key which exits the evaluation mode or in rare cases until the solution becomes unbounded whereupon it is deleted. While an attractor is being displayed the H R P and S keys can be used to change the way it is displayed without returning to the menu screen. The upper right-hand corner of the screen shows the number of kilobytes left to be evaluated in the file SA.DIC. When in the evaluation mode the program bypasses the calculation of the fractal dimension and Lyapunov exponent so that each case is displayed more quickly.

As you begin to accumulate a collection of favorites you will probably want to go back and find your favorites of the favorites. You merely need to rename the FAVORITE.DIC file to SA.DIC and evaluate them a second time. The attractors exhibited in this book were selected by this method after looking at about 100 000 cases. Since the FAVORITE.DIC file is ordinary ASCII text you can share your favorites with a friend who may have a different computer or operating system. You can easily e-mail the file to someone or upload it to a computer bulletin board or mainframe computer. Remember however that the programs in this book are copyrighted and are for your personal use. It is a violation of the copyright to share the programs with anyone else. You can now begin your own private collection of strange attractors artwork!

CHAPTER 6

Fields and Flows

In this chapter we consider equations whose iterates move gradually rather than abruptly from one place to another. Such equations are called differential equations and they are the basis for most dynamical systems that describe natural processes. The programming is a simple extension of what we have done before but the calculation requires more computing time. The attractors produced by differential equations consist of continuous lines whose weavings and waverings describe the trajectory and yield objects of considerable beauty.

6.1 Beam Me up Scotty!

The maps in the previous chapters have the property that successive iterates are usually at widely different positions on the attractor. The points dance around like fleas jumping on the back of a dog eventually but gradually visiting every allowed location. Most processes in nature don't occur that way but rather progress slowly and continuously from some initial condition through a succession of nearby intermediate states to the final condition.

If you take a trip across the country your trajectory through three-dimensional space (or even in four-dimensional space-time) is a continuous one-dimensional curve. Only in science fiction is Captain Kirk able to dematerialize at one position and rematerialize somewhere else without occupying a succession of intermediate positions. Most substances in nature obey a continuity equation which guarantees that if their quantity decreases at some position the decrease must be accompanied by a flow of the substance away from the position. Note that this is a stronger condition than a conservation law which requires only that the total quantity of the substance remains the same.

There is a relation between flows and maps. Imagine a fly trapped in a room and moving in a complicated random manner. Its trajectory is a one-dimensional curve that will eventually fill the entire room. However if you observe the fly with a strobe lamp that flashes periodically the trajectory will be a succession of dots with each dot separated from the previous dot by a significant distance. The dots also eventually fill the entire three-dimensional region but it takes longer for this to occur.

However if the fly's motion is chaotic rather than random neither the curve nor the dots will fill the room but rather they will lie on a strange attractor that occupies a negligible portion of the room. The attractor consisting of all the possible dots will often have a lower dimension than the attractor consisting of all the possible curves. Thus a map can be thought of as a crude description of a flow in which the intervening details of the motion are ignored.

It's easy to think of an object such as a fly or a human imbued with intelligence however limited moving by free will along a complicated trajectory. However inanimate objects such as astronomical bodies or sub-microscopic electrically charged particles can also execute complicated motions. They do so because they move through a space filled with gravitational or electromagnetic fields.

It is important to recognize that a field has no objective reality other than to describe mathematically the force on an object moving through it. When something is dropped it falls toward the Earth. It is a deeply philosophical question not answered very well by science how the object knows to move toward the Earth rather than in some other direction. We say that it is acted upon by the gravitational field of the Earth but this description however useful for calculating the motion begs the issue. Ultimately the laws of physics describe very accurately how things move but not very well why.

The equations that describe flows are of a different type than those that describe maps. They are called differential equations and they involve the rate of change of a quantity. We will consider only ordinary differential equations (ODEs) as distinguished from the partial differential equations (PDEs) that are used to describe the behavior of complicated objects like fluids whose state space is intrinsically infinite-dimensional. Dynamical systems described by ODEs involve only the time rate of change of the position of a point in state space whereas with PDEs the variables are quantities like density temperature and electric field that change in space as well as time. A wave is an example of a dynamical system described by a PDE.

Consider an object moving in the X-direction. Its speed is the rate of change of its position and we will denote this quantity by X' (pronounced "X prime"). It is the distance the object moves in a brief interval of time divided by the time interval. If you know some calculus you recognize this as the time-derivative of X usually denoted by dX/dt. The rate of change of position is what the speedometer on your car or the police radar reads. The rate of change of the speed is the acceleration. More properly we should call these quantities the time rate of change since quantities can also change in space. For example the spatial rate of change in altitude of a road is called its grade.

An object moving in three-dimensional space will have a constantly changing value not only of X but also of Y and Z. Furthermore X' Y' and Z' will usually depend on position (X Y and Z). For example a particle moving clockwise in a circle about the origin in the XY-plane is described by the following pair of differential equations:

X' = Y

(Eq. 6A)

Y' = -X

Such a set of equations describes at least approximately the motion of the Earth around the Sun. This type of regular motion is not chaotic and it does not lead to visually interesting strange attractors.

Some differential equations can be solved easily using calculus. For example Equation 6A has the solution

X = A sin(t + f)

(Eq. 6B)

Y = A cos(t + f)

which specifies the X and Y positions at any time t. The quantities A and f are constants that are determined from the initial conditions (the values of X and Y at t = 0). If you are interested only in the shape of the trajectory and not in where the object is along it at any particular time you can eliminate the t in Equations 6B to get a relation between X and Y

X2 + Y2 = A2 (Eq. 6C)

which is the equation for a circle of radius A centered on the origin (X = Y = 0).

Equation 6A also arises in a different context. Imagine an object moving back and forth in the X-direction perhaps attached to a spring that alternately stretches and compresses. Since Y is equal to X' we can associate Y with the velocity in the X-direction. The XY-plane then becomes the two-dimensional phase space for this one-dimensional motion and the trajectory in this plane is the phase-space trajectory. A circular phase-space trajectory is a characteristic of a one-dimensional simple harmonic oscillator such as a mass on a spring. Usually the phase-space trajectory is an ellipse just as the orbit of the Earth around the Sun is an ellipse but we can always measure Y in appropriate units or adjust the scale of the graph to change the ellipse into a circle.

With this interpretation the first of Equations 6A defines the velocity (Y) as the rate of change of position (X'). If you remember your physics the second of Equations 6A is Newton's second law (F = ma) in which the force F obeys Hooke's law for springs (F = - kX) and the acceleration a is the rate of change of velocity (Y'). It is interesting that the same set of differential equations with a change in the meaning of the variables can describe the motion of an object traveling in a circle or an object oscillating on the end of a spring. Equations 6A describe many other phenomena in nature such as the oscillations in an electrical circuit containing a capacitor and inductor.

A two-dimensional system of differential equations such as Equation 6A cannot exhibit chaos since the trajectory cannot cross itself. The most complicated bounded behavior is thus a simple closed loop corresponding to periodic motion. The reason the trajectory cannot cross itself is that every point in the XY-plane has associated with it a unique direction of flow and thus the trajectory must approach and leave every point in a single particular direction. If the orbit were to return to a point previously visited it would thereafter repeat what it did before. In two dimensions the orbit can do only one of three things--it can spiral into a fixed point approach a stable limit cycle or spiral off to infinity.

Trajectories may appear to cross if they come very close to a fixed point that is stable in one direction and unstable in another (called a saddle point or X-point because of its shape). Such a trajectory is called a separatrix since it separates regions with different flows. Trajectories approaching the fixed point on one side of the separatrix will veer off to the right and those approaching from the other side will veer off to the left. Such a separatrix exists upstream (and downstream) of an island in a river where two sticks placed side by side in the water end up going around opposite sides of the island. The island seems at first to attract the sticks and then to repel them at right angles as they approach it.

In three dimensions we have the possibility of an orbit wrapping around in a complicated manner like a ball of string never intersecting itself but producing a never-ending tangle. By contrast maps can be chaotic in one or two dimensions since the points jump from place to place with little danger of intersecting another point. Captain Kirk need not be concerned about a collision while being transported from one point to another. He only needs to worry about landing on top of a diabolical Klingon at his destination!

6.2 Professor Lorenz and Dr. Rössler

Although differential equations have been the mathematical basis for most descriptions of nature for hundreds of years almost no one suspected that the trajectories resulting from their solution could be a chaotic strange attractor. The history of the discovery of such solutions is interesting and bears retelling.

In the early 1960's Edward Lorenz a meteorologist at the Massachusetts Institute of Technology was developing models of atmospheric convection to be solved by a primitive computer that required about one second per iteration. His models involved a large number of differential equations and produced solutions that varied with time in a complicated manner not unlike the variation of the weather over long intervals of time. On one occasion he happened to restart one of his computer runs using numbers rounded to three digits rather than the six significant figures used by the computer.

For some time the solutions followed one another but after a while they began to depart and eventually they bore no relation to one another. He had discovered the sensitivity to initial conditions that is perhaps the most salient feature of chaos. He began simplifying his equations in an attempt to determine the minimum conditions necessary for this bizarre behavior. The result is the now famous Lorenz equations which represent the first example of a strange attractor arising from differential equations

X' = s(Y - X)

(Eq. 6D)

Y' = -XZ + rX - Y

Z' = XY - bZ

where s r and b are constants that Lorenz took to be s = 10 r = 28 and b = 8/3. Lorenz published his findings in 1963 in the Journal of the Atmospheric Sciences where they went largely unnoticed for the next decade. The title of his paper "Deterministic Nonperiodic Flow " is an apt description of what we now call "chaos."

Although the Lorenz equations were distilled from a model of atmospheric convection the trajectory in XYZ-space does not represent air currents in any literal way. Instead X corresponds to the size of the connective motion Y is proportional to the temperature difference between the ascending and descending fluids and Z is proportional to the deviation of the vertical temperature profile from a linear function. Nevertheless the behavior is reminiscent of a fluid with turbulent convection.

Since the Lorenz equations were proposed several phenomena have been found that are at least approximately modeled by them. Perhaps the simplest example is the thermosiphon. Imagine a continuous tube like a bicycle tube filled with a liquid and mounted vertically. If the bottom of the tube is heated and the top cooled a convection will ensue with the warm fluid rising and the cold fluid falling. The convection is equally likely to start in either direction. After it starts the circulation will continue in that direction a few times around the loop and then abruptly reverse.

In the 1970's other examples of chaotic differential equations began to be discovered. An important contribution was made in 1976 by Otto Rössler a non-practicing medical doctor in Germany. Rössler was interested in chaos in chemistry and in theoretical biology and he set about to find a system of equations even simpler than those of Lorenz that exhibited chaotic behavior. What he came up with are the now famous Rössler equations

X' = -(Y + Z)

(Eq. 6E)

Y' = X + aY

Z' = b + Z(X - g)

where a b and g are constants that Rössler took to be a = 0.2 b = 0.2 and g = 5.7. The Rössler equations constitute the simplest known example of a system of ordinary differential equations that exhibits chaos. They contain a single nonlinearity (ZX in the third equation). Rössler's original paper is also interesting because it contains a stereoscopic view of his strange attractor as well as the Lorenz attractor.

Until very recently the discovery of a new strange attractor was a cause to rush to publication. With the program in this book you can produce them by the thousands! Even today researchers tend to focus on a few well-known examples such as the Lorenz and Rössler attractors. An entire book has been written on the Lorenz attractor alone. Think of the libraries that could be filled by books describing your attractors in similar detail!

The Lorenz and Rössler attractors are shown in Figures 6-1 and 6-2 respectively albeit with slightly different values of the parameters than they used. These cases are known to have fractal dimensions slightly greater than 2.0. These examples are more important for their historical interest than for their visual appeal. If you have never seen these attractors in 3-D be sure to return to these cases and view them with the various display techniques after the program has been appropriately modified as described in the next section. The Lorenz attractor resembles the wings of a butterfly making it an appropriate emblem of chaos since the sensitivity to initial conditions is most dramatically illustrated by the butterfly effect.

QMCMMMWM

Figure 6-1. The Lorenz Attractor

QMMMMMIM

Figure 6-2. The Rössler Attractor

6.3 Finite Differences

Some differential equations such as Equations 6A can be solved exactly in a straightforward manner using calculus. However if a system of equations exhibits chaos no such solution is possible. The reason is that there is no mathematical function analogous to the sine and cosine that is capable of describing a strange attractor the way those functions describe a circle. The equations have to be solved by computer. We say that such solutions are numerical as opposed to analytical.

Unfortunately digital computers which are ideal for iterating maps are inherently incapable of exactly solving differential equations. The equations require that the solution advance slowly and smoothly. That is to say the successive iterates must differ by an infinitesimal amount and thus infinitely many iterations are required to make any progress. There are special analog computers designed for the task but they are not common or simple to program.

Books have been written on methods for the approximate numerical solution of differential equations and it is as much an art as a science. All the methods involve in one form or other a finite-difference approximation to the differential equation. Rather than take infinitesimal steps one advances in finite steps according to a prescription that attempts to minimize the inevitable errors. Fortunately for our purpose it is not necessary that the solutions be highly accurate and so we can use a simple procedure.

Perhaps the easiest and most transparent method for finding approximate solutions to differential equations is the Euler method. When this procedure is applied to the simple example of Equations 6A X and Y are advanced according to

Xn+1 = Xn + eYn

(Eq. 6F)

Yn+1 = Yn - eXn

where e is the time step that ideally should be negligibly small but in reality is made as large as possible to reduce the number of iterations required to advance the solution by a substantial distance along the trajectory. You see that the Euler method is just another example of an iterated map in which successive iterates are near one another. It is perhaps the least accurate method for solving differential equations and it is easily improved upon. However for most of our purposes the Euler method is adequate. Furthermore it is simple to modify the program to solve differential equations by this method. The necessary changes are shown in the listing PROG21.

PROG21 Changes Required in PROG20 to Solve Differential Equations by the Euler Method

1000 REM ODE SEARCH

1070 D% = 3                     'Dimension of system
1080 EPS = .1                   'Step size for ODE
1090 ODE% = 1                   'System is ODE
1990 IF ODE% = 1 THEN XN(I%) = XY(I%) + EPS * XN(I%)
2660    CODE$ = CHR$(59 + 4 * D% + O% + 8 * ODE%)
3050    IF ODE% = 1 THEN L = L / EPS
3660 IF ODE% = 1 THEN D% = D% + 2
3680 IF Q$ = "D" THEN D% = 1 + (D% MOD 6): T% = 1
3700 IF D% > 4 THEN ODE% = 1: D% = D% - 2 ELSE ODE% = 0
4300    PRINT TAB(27); "D: System is"; STR$(D%); "-D polynomial ";
4310    IF ODE% = 1 THEN PRINT "ODE" ELSE PRINT "map"
4730 IF D% > 4 THEN D% = D% - 2: ODE% = 1 ELSE ODE% = 0

In PROG21 the value of e has been taken as 0.1 and the three-dimensional equations are taken to be polynomials up to fifth order with coefficients chosen by analogy to the three-dimensional polynomial maps previously described. We don't consider differential equations in less than three dimensions since they cannot have chaotic solutions. The second-order through fifth-order equations are coded with the first letters Q R S and T respectively.

If e is sufficiently small its value should not affect whether a system is chaotic or the general appearance of the attractor but it will certainly change the trajectory on the attractor. Just as a chaotic trajectory is sensitive to initial conditions it is also sensitive to the approximations used to calculate it. Unfortunately a value of e = 0.1 is not sufficiently small and many of the resulting attractors would disappear or change their appearance if e were reduced. Conversely other attractors would emerge for smaller values of e. Fortunately for our purposes it is not necessary that the solutions be even qualitatively correct. You should be forewarned that reducing e will have unpredictable effects on the attractors and that the computation time will increase.

The Lyapunov exponent is calculated as with the corresponding maps except that it is divided by e so that its units are bits per second rather than bits per iteration since each iteration advances the solution by e seconds. It is customary to express the Lyapunov exponent in this way for differential equations since the step size depends upon the numerical approximation that is being used whereas the divergence of the trajectories per unit time is an intrinsic property of the differential equations.

Sample attractors produced by three-dimensional ordinary differential equations projected onto the XY-plane are shown in Figures 6-3 through 6-6.

QRREQDTW

Figure 6-3. Projection of Three-Dimensional Quadratic ODE

RMGOLOGU

Figure 6-4. Projection of Three-Dimensional Cubic ODE

SSKBHUTG

Figure 6-5. Projection of Three-Dimensional Quartic ODE

TLFBEBGP

Figure 6-6. Projection of Three-Dimensional Quintic ODE

The preceding figures are like time-exposed photographs of the shadow on the wall of a fly moving chaotically in a room. However because of the finite difference approximation used to solve the equation of motion you must imagine that the fly is illuminated by a strobe lamp that flashes rapidly. The trajectory thus consists of a large number of closely spaced dots. The separation of the dots provides a measure of the accuracy of the solution. Although some of the cases produce apparently continuous trajectories others more nearly resemble the maps of the previous chapters. You might prefer to alter the program so that the dots are connected by lines.

Another consequence of dealing with differential equations is that many iterations are required for the solution to settle onto the attractor. Since we use the same criterion for the number of iterations as we did for the maps a significant number (perhaps 20 percent) of the attractors found in a random search are not chaotic and a few are even unbounded. When you evaluate the attractors found by the search you will recognize these cases by the way they eventually settle onto a simple closed curve that is visually indistinguishable from a limit cycle spiral into a fixed point or leave the screen. You will also notice a few cases that consist of isolated islands with no bridge connecting them such as the one in Figure 6-3. You can be sure these are not true flows since such discontinuities are impossible in the trajectories that arise from the solution of differential equations.

There is no reason to limit the display of attractors arising from differential equations to projections onto a plane. All the display techniques developed in Chapter 4 for three-dimensional maps are also available here. Figures 6-7 through 6-22 and Plates 23 and 24 show a selection of such examples.

QDEIXNUK

Figure 6-7. Three-Dimensional Quadratic ODE with Shadows

RKCDFPFV

Figure 6-8. Three-Dimensional Cubic ODE with Shadows

SUISTHIX

Figure 6-9. Three-Dimensional Quartic ODE with Shadows

TQHTPBVX

Figure 6-10. Three-Dimensional Quintic ODE with Shadows

QLOTRIDM

Figure 6-11. Three-Dimensional Quadratic ODE with Contour Bands

RHNKDRSE

Figure 6-12. Three-Dimensional Cubic ODE with Contour Bands

SAQKAETC

Figure 6-13. Three-Dimensional Quartic ODE with Contour Bands

TIKDSMEK

Figure 6-14. Three-Dimensional Quintic ODE with Contour Bands

QEDWIVON

Figure 6-15. Stereo Pair of Three-Dimensional Quadratic ODE

RFEJMURN

Figure 6-16. Stereo Pair of Three-Dimensional Cubic ODE

SHBNLCIM

Figure 6-17. Three-Dimensional Quartic ODE with Contour Bands

TBEIESEQ

Figure 6-18. Three-Dimensional Quintic ODE with Contour Bands

QUCTPNYF

Figure 6-19. Slices of Three-Dimensional Quadratic ODE

RNMKSFEK

Figure 6-20. Slices of Three-Dimensional Cubic ODE

SQFECQRR

Figure 6-21. Slices of Three-Dimensional Quartic ODE

THBDSQAR

Figure 6-22. Slices of Three-Dimensional Quintic ODE

6.4 Flows in Four Dimensions

Flows like maps can be embedded in spaces of arbitrary dimension. Four-dimensional flows are hard to visualize but pose no difficulty for the computer to calculate. Buried in the listing PROG21 is the capability for calculating four-dimensional flows. You only need to press the D key to access the four-dimensional ODE's. All the techniques previously developed for displaying four-dimensional maps are available. The quadratic cubic quartic and quintic equations are coded with the first letters U V W and X respectively. Figures 6-23 through 6-38 and Plates 25 through 30 show a selection of strange attractors arising from four-dimensional ordinary differential equations with polynomial terms.

UJKCAWNX

Figure 6-23. Projection of Four-Dimensional Quadratic ODE

VGKEMAGQ

Figure 6-24. Projection of Four-Dimensional Cubic ODE

WTQRFNAV

Figure 6-25. Projection of Four-Dimensional Quartic ODE

XNOYBECR

Figure 6-26. Projection of Four-Dimensional Quintic ODE

UHUPENUE

Figure 6-27. Four-Dimensional Quadratic ODE with Shadow Bands

VKCLEUHO

Figure 6-28. Four-Dimensional Cubic ODE with Shadow Bands

WKDEDPGP

Figure 6-29. Four-Dimensional Quartic ODE with Shadow Bands

XTPXAPOK

Figure 6-30. Four-Dimensional Quintic ODE with Shadow Bands

UGLICIRJ

Figure 6-31. Four-Dimensional Quadratic ODE with Stereo Bands

VHTQNWHD

Figure 6-32. Four-Dimensional Cubic ODE with Stereo Bands

WJCUJYON

Figure 6-33. Four-Dimensional Quartic ODE with Stereo Bands

XJUSDQKD

Figure 6-34. Four-Dimensional Quintic ODE with Stereo Bands

Figure 6-35. Four-Dimensional Quadratic ODE with Sliced Bands

VIPHNKSO

Figure 6-36. Four-Dimensional Cubic ODE with Sliced Bands

WNHRNECE

Figure 6-37. Four-Dimensional Quartic ODE with Sliced Bands

XPOOCSEN

Figure 6-38. Four-Dimensional Quintic ODE with Sliced Bands

6.5 Strange Attractors that Aren't

In Section 3.8 we discussed chaotic orbits that don't approach an attractor (Strange Attractors that Don't). Here we consider non-chaotic orbits that approach attractors that aren't strange. These attractors are not fractals. They have dimensions that are integers such as 0 1 2 or 3. Some of them are beautiful and thus they are worth displaying even if they are technically outside the scope of this book.

Such attractors can arise from maps as well as from differential equations. They don't require high embedding dimensions although the dimension of the attractor will always be at least one less than the dimension of the embedding space. Thus some of the examples are taken from equations described in earlier chapters.

The simplest non-chaotic attractor is a point attractor. Suppose we modified Equations 6A so that the solution was not a circle but an inward spiral. One way to do this is as follows:

X' = Y - bX

(Eq. 6G)

Y' = -X - bY

You can think of the coefficient b as a measure of the friction that eventually brings the trajectory to rest at the origin (X = Y = 0) in phase space. If b is zero (frictionless) the orbit is a circle. Negative values of b (anti-friction) cause the solution to spiral outward approaching infinity. This case corresponds to a point repellor at the origin.

The two occurrences of b in Equations 6G need not have the same value or even the same sign. In such a case the orbit spirals in or out but not in a perfectly symmetrical manner. Many physical processes have b = 0 in one of the equations. If b is close to zero it doesn't matter much in which equation it appears.

Small positive values of b cause the radius of the circle to decrease slowly approaching what is called a spiral-point or focal-point attractor or simply a focus. Larger positive values of b cause the radius to decrease more rapidly. With very large values of b there is little circulation around the point and the trajectory is more nearly radial toward what is called a radial-point or nodal-point attractor or simply a node. The boundary between the two cases corresponds to critical damping in an oscillator. In either case the resulting attractor is a point at the origin with a dimension of zero. A code that produces a point attractor (with b = 1) is QMLM3NM5LM3LM14.

The above case has a negative value of the largest Lyapunov exponent (L = - 0.14) and it produces a single dot on the screen. More interesting cases can occur because the program assumes the trajectory is on the attractor after 1000 iterations. For trajectories that approach the attractor very slowly there can be interesting behavior after the thousandth iteration and before the fixed point is reached. Such slowly attracting fixed points have Lyapunov exponents very close to zero. The search can be expanded to include them as well as other non-chaotic attractors by changing the .005 in line 2480 of the program to -.005 for example. Then attractors with Lyapunov exponents near zero but which have not settled to a fixed point after NMAX iterations will be included in the file SA.DIC. If you prefer you can collect them in a separate file TORUS.DIC by changing line 4910 of PROG21 to

4910 IF L > .005 THEN OPEN "SA.DIC" FOR APPEND AS #1 ELSE OPEN "TORUS.DIC" FOR APPEND AS #1

Several such cases are shown in Figures 6-39 through 6-42. Figure 6-39 shows a spiral-point attractor. Figure 6-40 shows what appears to be a radial-point attractor with several different initial conditions but which is really a spiral-point attractor with successive iterates that move rapidly around the fixed point. This phenomenon is called aliasing and it is most easily detected by connecting temporally successive points with continuous lines. Figures 6-41 and 6-42 show cases where the rate of circulation around the fixed point changes significantly as the fixed point is approached.

TNGLFVWP

Figure 6-39. Trajectory Approaching a Spiral-Point Attractor

QGNSFHVS

Figure 6-40. Trajectory Emulating a Radial-Point Attractor

QAVXVXDX

Figure 6-41. Trajectory Approaching a Point Attractor

RFOWMYQV

Figure 6-42. Trajectory Approaching a Point Attractor

The point attractor is the simplest type of non-chaotic attractor. It has a dimension of zero. An attractor can also have a dimension of one which is a line. Such attractors are limit cycles. They correspond to systems that settle into a periodic or cyclic behavior. Consequently such attractors are also called cyclic attractors.

The simplest differential equations that produce a cyclic trajectory are Equations 6A. The resulting orbit is a circle in the XY-plane. This case is not an attractor however because every initial condition produces a circular trajectory whose radius is the distance of the initial point from the origin. There is no unique circle to which nearby orbits are attracted and there is no basin of attraction. Furthermore if you attempt to display the trajectory using a code such as QM5NM5LM18 you will find that the orbit is unbounded and spirals outward as if there were a point repellor at the origin. The reason is that our iteration scheme for approximating the solution of the differential equations is not exact. The errors compound and eventually cause the orbit to be lost.

The simple undamped (frictionless) oscillator is said to be structurally unstable because an arbitrarily small perturbation to the equation (such as using the finite difference approximation of Equation 6F) changes the structure of the solution from a closed loop to a never-ending spiral. Note the distinction between an unstable equation in which a small modification of the equation causes a large change in the solution and an unstable solution in which a small variation of the initial condition away from the equilibrium value causes the solution to move ever farther from equilibrium.

To produce a true limit cycle that is structurally stable we need a system of equations whose solutions spiral outward from initial conditions in the interior and spiral inward from initial conditions on the exterior of the attractor. A suitable set of such equations is the following:

X' = Y + (1 - X2 - Y2)X

(Eq. 6H)

Y' = -X + (1 - X2 - Y2)Y

The quantity (1 - X2 - Y2) plays the role of -b in Equations 6G. Whenever the trajectory lies inside the circle of radius one it spirals outward and whenever it lies outside the circle of radius one it spirals inward. Thus the limit cycle is defined by the circle X2 + Y2 = 1. There is a point repellor at the origin and the basin of attraction is the entire XY-plane. A code that produces such a stable limit cycle except with a smaller radius is RMNMAM3AM3NM9LM2AM6NMAM26.

A limit cycle may be either stable or unstable just like a fixed point. With an unstable limit cycle nearby orbits move progressively farther from the limit cycle. An unstable limit cycle can be identified in an invertable map or system of ordinary differential equations by running time backwards in which case the limit cycle becomes stable and attracts rather than repels nearby orbits.

A slightly simpler version of Equations 6H that produces a stable limit cycle although not a symmetrical one is the following:

X' = Y

(Eq. 6I)

Y' = -X + (1 - X2)Y

This system is called the Van der Pol equation and it was first used to model vacuum-tube oscillator circuits but it has been used in other applications such as the modeling of pulsating stars called Cephids. A code for the Van der Pol equation is RM11OM9KM2KM6OM28.

Such limit cycles are characterized by a dimension of one and a Lyapunov exponent of zero. The dimension as approximated by the program will usually not be exactly 1.0 for the reasons discussed in Section 3.4. The Lyapunov exponent is a much better criterion for identifying limit cycles. In a two-dimensional embedding space as in the above examples there are two Lyapunov exponents. The smaller (more negative) of them is the rate at which trajectories with different initial conditions approach the attractor. The larger exponent (the one calculated by the program) is the rate at which two nearby points on the limit cycle separate. For the attractor to be non-chaotic (not strange) this exponent must be zero.

In two dimensions the limit cycles cannot cross and so the most complicated shapes are simple distorted loops. In three or more dimensions they can wrap around in a complicated tangle like a ball of string but without ends. Figures 6-43 through 6-50 show a collection of visually interesting limit cycles. They are plotted as stereo pairs so that you can see how the trajectories pass beneath one another.

As you examine the figures you will note that some of the limit cycles such as the one in Figure 6-43 form knots. You cannot straighten them out into circular loops. By contrast Figure 6-47 is unknotted. This knottedness or helicity is an important topological property of an attractor. Some processes in nature tend to conserve helicity just as mechanical energy is conserved in frictionless motion. Thus when some parameter of the system is changed the limit cycle may change its size and shape but in such a way that it always links itself in the same way. An example is a magnetic field line in a turbulent conducting fluid such as a plasma of electrically charged particles.

For many of the limit cycles exhibited here it is very hard to tell whether they are knotted. Even when they appear to be knotted it is hard to tell whether two cases are knotted in the same way. Such patterns might provide a useful psychological test for one's spatial acuity since they require both depth perception and a mental dexterity to visualize their shape when untangled as much as possible. See which of the limit cycles in the figures you think are knotted.

INSIUROM

Figure 6-43. Limit Cycle from Three-Dimensional Quadratic Map

QDIPESXN

Figure 6-44. Limit Cycle from Three-Dimensional Quadratic ODE

QQDKWCVB

Figure 6-45. Limit Cycle from Three-Dimensional Quadratic ODE

QUKURGZH

Figure 6-46. Limit Cycle from Three-Dimensional Quadratic ODE

RSCVKBOZ

Figure 6-47. Limit Cycle from Three-Dimensional Cubic ODE

TFRKCBHS

Figure 6-48. Limit Cycle from Three-Dimensional Quintic ODE

UUJBACFH

Figure 6-49. Limit Cycle from Four-Dimensional Quadratic ODE

WNFIDCJH

Figure 6-50. Limit Cycle from Four-Dimensional Quartic ODE

6.6 Doughnuts and Coffee Cups

Non-chaotic attractors can be points or lines. They can also be surfaces. Surfaces are two-dimensional manifolds. Perhaps the simplest set of equations whose solution is a trajectory that fills a surface is the following:

X' = Y

Y' = -X

(Eq. 6J)

Z' = aW

W' = -Z

You will recognize the first two equations as the same as Equations 6A that produce a circle in the XY-plane. The second two equations produce an ellipse in the ZW-plane. The two motions are uncoupled (X and Y don't depend on Z or W; Z and W don't depend on X or Y). The parameter a is the square of the angular frequency of the second motion. If the square root of a is a rational number (a ratio of two integers) the trajectory is a closed one-dimensional loop in the four-dimensional embedding space.

If the square root of a is irrational the trajectory fills a two-dimensional toroidal surface (called a 2-torus). The trajectory winds endlessly around the surface of a dougnut never intersecting itself. In such a case we say the frequencies (the number of transits per second the long way around and the short way around) are incommensurate and that the trajectory is quasi-periodic. The sequence never repeats but it is not chaotic. It is sometimes difficult to distinguish between quasi-periodic and chaotic behavior.

A useful tool for distinguishing between a quasi-periodic and a chaotic attractor is the power spectrum of the time series which will have sharp peaks at discrete frequencies for quasi-periodic trajectories but a broad (continuous) spectrum for chaotic trajectories. The power spectrum contains about half of the information required to reconstruct the trajectory; the frequency information is present but the phase information is lost. Nevertheless the power spectrum serves as a kind of fingerprint that is very useful in categorizing attractors.

Equations 6J have the same problems as Equations 6A. They don't represent an attractor since nearly all initial conditions produce different tori. Furthermore the tori produced in this way are structurally unstable just like the circles of Equations 6A. These difficulties can be circumvented by using instead an extension of Equations 6H to produce two uncoupled limit cycles as follows:

X' = Y + (1 - X2 - Y2)X

Y' = -X + (1 - X2 - Y2)Y

(Eq. 6K)

Z = aW + (1 - Z2 - W2)Z

W = -Z + (1 - Z2 - W2)W

A value of a = 2 provides an acceptable irrational frequency ratio since the square root of 2 cannot be represented as a ratio of two integers. The corresponding trajectory can be generated using the code VMNMAM4AM7NM19LM2AM11NMAM42NMAM2AOM28LM2AM2NMA. A rotated version of the 2-torus in which one loop is in the XZ-plane and the other is in the YW-plane is produced by the code VMNMAM8AM13NM24NMAM6AM6OM3LM3AM20NMAM22LM3AM11NMA.

Two uncoupled limit cycles lie on a torus that is attractive but it is not technically an attractor; it is called an invariant manifold. For an object to be an attractor it must not only attract nearby trajectories but most trajectories on it must wander all over it in which case we say the set is transitive and the orbits are ergodic. Ergodic orbits produce mixing which means that an orbit starting from anywhere on the attractor will eventually come arbitrarily close to every other point on the attractor. Mixing ensures that an attractor cannot be split into two different attractors although the attractor need not be connected. Figure 5-11 shows an attractor that is not connected. Thus not all attractive tori are attractors just as not all attractors are tori.

Tori can be identified in the computer search by a Lyapunov exponent close to zero and a dimension well above one. It is easy to distinguish them visually from limit cycles which can also have Lyapunov exponents close to zero but which resemble lines rather than surfaces. A selection of tori projected onto the XY-plane is shown in Figures 6-51 through 6-60.

REZCDWWW

Figure 6-51. Torus from Three-Dimensional Cubic ODE

RXEGQFBY

Figure 6-52. Torus from Three-Dimensional Cubic ODE

UELMCAPH

Figure 6-53. Torus from Four-Dimensional Quadratic ODE

UFYVWDQL

Figure 6-54. Torus from Four-Dimensional Quadratic ODE

UHLDVFVI

Figure 6-55. Torus from Four-Dimensional Quadratic ODE

UIDVAPPB

Figure 6-56. Torus from Four-Dimensional Quadratic ODE

URDNTMWH

Figure 6-57. Torus from Four-Dimensional Quadratic ODE

UWBDELAT

Figure 6-58. Torus from Four-Dimensional Quadratic ODE

VFFSCILF

Figure 6-59. Torus from Four-Dimensional Cubic ODE

WJUKEUQU

Figure 6-60. Torus from Four-Dimensional Quartic ODE

Most of the attractors shown in the figures look like tori in the sense that you can see or imagine the hole in the doughnut. However it is important to understand that just as not all limit cycles are circles not all 2-tori look like doughnuts. They are topologically equivalent in the sense that there is a "rubber-sheet" deformation (called a homeomorphism) that maps them into a doughnut. A coffee cup for example is topologically equivalent to a doughnut as long as the handle is unbroken.

Those cases that are not obviously equivalent to a simple torus are distorted by the fact that they are viewed projected onto the XY-plane or because they are rotated at an awkward angle. Also note that most of these tori are embedded in a four-dimensional space and thus it is even more difficult to grasp their shape from a two-dimensional projection. You might wish to display them using some of the advanced visualization techniques provided by the program.

It is possible though difficult to produce a 3-torus in a four-dimensional embedding space. A 3-torus is a generalization of a 2-torus. It is hard to visualize. It involves three mutually incommensurate frequencies. It is characterized by a dimension of three and a largest Lyapunov exponent of zero. Some of the attractors in the figures seem to be 3-tori according to their calculated dimension. However the calculation is not sufficiently precise to distinguish unambiguously between a 2-torus and a 3-torus. It is necessary to search embedding dimensions greater than four to have a good chance of finding 3-tori.

Dynamical systems whose trajectories lie on a 3-torus or other hypertori of even higher dimensions are difficult to observe in nature. The reason is that such attractors can be perturbed by an arbitrarily small change to the system that causes them to become strange attractors. According to Peixoto's theorem (which strictly applies only to compact orientable manifolds) 2-tori tend to be structurally stable while 3-tori and higher are structurally unstable. Thus it appears that complicated deterministic systems that exhibit nontrivial behavior are well represented by the strange attractors that constitute the subject of this book.

CHAPTER 7

Further Fascinating Functions

For a system of equations to exhibit chaos the equations must contain at least one nonlinear term that is a term that is not simply proportional to one of the variables. In all the preceding examples the nonlinearity involved simple polynomials. Such polynomials are capable of modeling an enormous variety of physical phenomena. Virtually all nonlinear functions can be approximated by polynomials with sufficiently many terms. However by limiting the polynomials to fifth order we have missed many interesting possibilities. In this chapter we will examine a few of these possibilities and suggest others that you might wish to explore on your own.

7.1 Steps and Tents

Perhaps the simplest nonlinear function is the absolute value which is denoted by |X| and programmed in BASIC with the command ABS(X). The absolute value of X is the magnitude of X without regard to its sign. For example if X is -6 then |X| is 6. It is a nonlinear function because a graph of |X| versus X is a V-shaped curve with its notch at the origin rather than a straight line as would result if |X| were proportional to X. By adding linear terms the V can be rotated to resemble an L or a staircase step. Computers can evaluate ABS(X) very quickly since they only need to discard the sign.

An example of a one-dimensional chaotic map that involves |X| is the tent map so called because its graph is an inverted V. A tent map that maps the interval -1 to 1 back onto itself is

Xn+1 = 1 - 2|Xn| (Eq. 7A)

The behavior of Equation 7A is very similar to the behavior of the logistic equation (Equation 1C) with R = 4 which maps the interval 0 to 1 back onto itself. A mapping that returns a set onto itself is called an endomorphism.

Since one-dimensional maps tend not to be very interesting visually we will generalize Equation 7A to two dimensions as follows:

Xn+1 = a1 + a2Xn + a3Yn + a4|Xn| + a5|Yn|

(Eq. 7B)

Yn+1 = a6 + a7Xn + a8Yn + a9|Xn| + a10|Yn|

This form is analogous to the general two-dimensional quadratic map in Equations 3B.

To make things a little more interesting we can add two more dimensions (Z and W) to take advantage of the visualization techniques that have been previously developed. However to keep things simple we will demand that X and Y don't depend on Z or W. They are just along for the ride so to speak. The dynamical behavior will be determined only by X and Y. We can choose any convenient equation for Z and for W. One possibility is to evaluate Z from X and Y according to

Zn+1 = Xn2 + Yn2 (Eq. 7C)

Thus Z is the square of the distance of the previous iterate from the origin. You might want to experiment with other forms.

For the fourth dimension (W) we will do something completely different. We will arrange for W to increase linearly with the iteration number. Thus W becomes the time coordinate in four-dimensional space-time.

The program modifications required to extend the computer search to such cases are shown in the listing PROG22. Codes for this case begin with the letter Y.

PROG22 Changes Required in PROG21 to Search for Special Functions of the Y Type

1000 REM SPECIAL FUNCTION SEARCH (Steps and Tents)

1090 ODE% = 2                   'System is special function Y
1710 IF ODE% > 1 THEN GOSUB 6200: GOTO 2020     'Special function
2670    IF ODE% > 1 THEN CODE$ = CHR$(87 + ODE%)
3650 IF ODE% > 1 THEN D% = ODE% + 5
3680 IF Q$ = "D" THEN D% = 1 + (D% MOD 7): T% = 1
3690 IF D% > 6 THEN ODE% = D% - 5: D% = 4: GOTO 3710
4290    IF ODE% > 1 THEN PRINT TAB(27); "D: System is 4-D special map "; CHR$(87 + ODE%); " ": GOTO 4320
4720 IF D% > 6 THEN ODE% = ASC(LEFT$(CODE$
1)) - 87: D% = 4: GOSUB 6200: GOTO 4770
6200 REM Special function definitions
6210 ZNEW = X * X + Y * Y       'Default 3rd and 4th dimension
6220 WNEW = (N - 100) / 900: IF N > 1000 THEN WNEW = (N - 1000) / (NMAX - 1000)
6230 IF ODE% <> 2 THEN GOTO 6270
6240    M% = 10
6250    XNEW = A(1) + A(2) * X + A(3) * Y + A(4) * ABS(X) + A(5) * ABS(Y)
6260    YNEW = A(6) + A(7) * X + A(8) * Y + A(9) * ABS(X) + A(10) * ABS(Y)
6270 RETURN

Examples of attractors produced by PROG22 are shown in Figures 7-1 through 7-8. They are displayed as projections onto the XY-plane to let you observe the higher dimensional representations for the first time on your computer screen. Note that these attractors differ from the cases produced by polynomials in that they tend to have sharp angular corners. The one in Figure 7-3 is not an attractor but is an example of an area-preserving (Hamiltonian) system sometimes called the gingerbread man because of its shape.

YBTDNPHI

Figure 7-1. Four-Dimensional Special Map Y

YCHTPNOB

Figure 7-2. Four-Dimensional Special Map Y

YCMCCMMW

Figure 7-3. Four-Dimensional Special Map Y (Gingerbread Man)

YDGPCXGX

Figure 7-4. Four-Dimensional Special Map Y

YKXILQOR

Figure 7-5. Four-Dimensional Special Map Y

YOBLSVUL

Figure 7-6. Four-Dimensional Special Map Y

YPSRTGND

Figure 7-7. Four-Dimensional Special Map Y

YYEQLKXB

Figure 7-8. Four-Dimensional Special Map Y

7.2 ANDs and ORs

A very different type of nonlinear map can be produced using logical (Boolean) operations to manipulate the individual bits of the binary numbers that represent the variables. This is best done after rounding the variables to the nearest integer using the BASIC CINT function. As a variation you could use the FIX or INT functions which truncate rather than round. Most versions of BASIC automatically apply the CINT function before performing logical operations on numbers that are not integers. The conversion of a non-integer to an integer is itself a nonlinear operation since the graph of CINT(X) versus X would resemble a staircase.

The basic logical operators are AND and OR. If you are not sure what these operations mean your BASIC manual is a good reference. The operation X AND Y produces a new number whose bits are 1 if the corresponding bits of X and Y are both 1 and 0 otherwise. The operation X OR Y produces a new number whose bits are 1 if either (or both) of the corresponding bits of X or Y are 1 and 0 otherwise. This is also called the inclusive OR to distinguish it from the exclusive OR (XOR) which produces a number whose bits are 1 if either (but not both) of the corresponding bits of X or Y are 1 and 0 otherwise.

A general two-dimensional system of equations that includes the AND and OR operators is the following:

Xn+1 = a1 + a2Xn + a3Yn + a4Xn AND a5Yn + a6Xn OR a7Yn

(Eq. 7D)

Yn+1 = a8 + a9Xn + a10Yn + a11Xn AND a12Yn + a13Xn OR a14Yn

The third and fourth dimensions are determined in the same way as in the previous section.

The program modifications required to extend the computer search to such cases are shown in the listing PROG23. Codes for this case begin with the letter Z.

PROG23 Changes Required in PROG22 to Search for Special Functions of the Z Type

1000 REM SPECIAL FUNCTION SEARCH (ANDs and ORs)

1090 ODE% = 3                   'System is special function Z
3680 IF Q$ = "D" THEN D% = 1 + (D% MOD 8): T% = 1
6270 IF ODE% <> 3 THEN GOTO 6310
6280    M% = 14
6290    XNEW = A(1) + A(2) * X + A(3) * Y + (CINT(A(4) * X) AND CINT(A(5) * Y)) + (CINT(A(6) * X) OR CINT(A(7) * Y))
6300    YNEW = A(8) + A(9) * X + A(10) * Y + (CINT(A(11) * X) AND CINT(A(12) * Y)) + (CINT(A(13) * X) OR CINT(A(14) * Y))
6310 RETURN

Examples of attractors produced by PROG23 are shown in Figures 7-9 through 7-16. Most of the attractors produced in this way have a streaked or checkered appearance arising presumably from rounding the variables to integers before performing the logical operations. The ones shown in the figures tend to be the exceptions.

ZHXMLJJR

Figure 7-9. Four-Dimensional Special Map Z

ZIDYKMPG

Figure 7-10. Four-Dimensional Special Map Z

ZMEQGDPO

Figure 7-11. Four-Dimensional Special Map Z

ZOCYAXYF

Figure 7-12. Four-Dimensional Special Map Z

ZOFFLRTE

Figure 7-13. Four-Dimensional Special Map Z

ZTXNODBP

Figure 7-14. Four-Dimensional Special Map Z

ZVPTKKHL

Figure 7-15. Four-Dimensional Special Map Z

ZXQUMEEG

Figure 7-16. Four-Dimensional Special Map Z

7.3 Roots and Powers

Polynomial maps involve powers of the variables that are small positive integers such as the square (2) and the cube (3). Polynomials exclude nonlinearities such as the square root or the reciprocal of the variables. Roots and powers are mathematically equivalent except for the size of the exponent. The square root of X can be written as X0.5 and the reciprocal of X can be written as 1/X or as X-1. It is interesting to examine strange attractors that involve fractional and negative powers.

A general two-dimensional system of equations that involves arbitrary powers is the following:

Xn+1 = a1 + a2Xn + a3Yn + a4|Xn|a5 + a6|Yn|a7

(Eq. 7E)

Yn+1 = a8 + a9Xn + a10Yn + a11|Xn|a12 + a13|Yn|a14

The absolute values are needed because BASIC cannot take a root of a negative number. The result would have an imaginary component. Note that if all the exponents happen to be +1 Equations 7E are equivalent to Equations 7B. The third and fourth dimensions are determined in the same way as in Section 7.1.

The program modifications required to extend the computer search to such cases are shown in the listing PROG24. Since we have exhausted the capital letter codes it is necessary to invent some new ones. We will continue using the standard ASCII characters beyond Z as shown in Table 2.1. Thus the codes for this case will begin with the left bracket ([) which is ASCII 91.

PROG24 Changes Required in PROG23 to Search for Special Functions of the [ Type

1000 REM SPECIAL FUNCTION SEARCH (Roots and Powers)

1090 ODE% = 4                   'System is special function [
3680 IF Q$ = "D" THEN D% = 1 + (D% MOD 9): T% = 1
6310 IF ODE% <> 4 THEN GOTO 6350
6320    M% = 14
6330    XNEW = A(1) + A(2) * X + A(3) * Y + A(4) * ABS(X) ^ A(5) + A(6) * ABS(Y) ^ A(7)
6340    YNEW = A(8) + A(9) * X + A(10) * Y + A(11) * ABS(X) ^ A(12) + A(13) * ABS(Y) ^ A(14)
6350 RETURN

Examples of attractors produced by PROG24 are shown in Figures 7-17 through 7-24. These attractors are localized mostly to a small region of the XY-plane with tentacles that stretch out to large distances. If any of the exponents are negative and the attractor intersects the line along which the respective variable is zero a point on the line will map to infinity. However large values are visited infrequently by the orbit and so many iterations are required to determine that the orbit is unbounded. For this reason most of the attractors in the figures have holes in their interior where the orbit is precluded from coming too close to the origin (X = Y = 0).

You may become frustrated to see a beautiful attractor develop for thousands of iterations and then to have the orbit escape. Such behavior is called a crisis or transient chaos not to be confused with a catastrophe. For example the logistic equation with R slightly greater than 4.0 is chaotic for many iterations until an iterate happens to exceed 1.0 whereupon the orbit abruptly moves off toward infinity. By contrast a catastrophe occurs when the solution undergoes a qualitative change at some critical value of a control parameter.

Related to a crisis is another phenomenon called intermittency. At certain values of the control parameters a system will exhibit periodic behavior for many cycles and then will suddenly become chaotic for a while before settling back into periodic behavior. Classic examples of intermittency occur in the logistic equation at about R = 3.82812 and in the Lorenz equations at about r = 166.2. Intermittency has been observed in many natural systems and it is a bane to those who try to make predictions. It is possible that the solar system is intermittently chaotic or even that a crisis could occur leading to a complete loss of its stability perhaps precipitated by a rare conjunction of a planet with a large asteroid or comet.

Those solutions of Equations 7E that remain bounded tend to have a wispy appearance and to go beyond the frame of the figure because of the occasional large excursions. Artistically this feature gives the attractors a sense of being connected to the surrounding world rather than being isolated objects suspended in a void. If you frame these cases a surrounding mat is desirable to provide the illusion that they are being viewed through a window.

[ADXWOPT

Figure 7-17. Four-Dimensional Special Map [

[BLWBPPF

Figure 7-18. Four-Dimensional Special Map [

[CBJGRMD

Figure 7-19. Four-Dimensional Special Map [

[EQBRGLH

Figure 7-20. Four-Dimensional Special Map [

[LDTQTTL

Figure 7-21. Four-Dimensional Special Map [

[TTKFUVI

Figure 7-22. Four-Dimensional Special Map [

[TTOLLRO

Figure 7-23. Four-Dimensional Special Map [

[WJSFLSK

Figure 7-24. Four-Dimensional Special Map [

7.4 Sines and Cosines

One of the most common nonlinear function is the sine and its complement the cosine. The sine and cosine can be approximated by polynomials as follows:

sin X = X - X3/6 + X5 /120 - X7/5040 + ...

(Eq. 7F)

cos X = 1 - X2/2 + X4/24 - X6/720 + ...

When the argument X is small the approximations are very accurate using only a few terms in the expansion. The denominator of each term is the factorial of the exponent of that term. For example the factorial of five (written 5!) is equal to 5 x 4 x 3 x 2 x 1 = 120. When X is large many terms are required. In such a case we would expect to observe dynamics different from those produced by the fifth-order polynomials previously examined.

A general two-dimensional system of equations whose nonlinearity is restricted to the sine function is the following:

Xn+1 = a1 + a2Xn + a3Yn + a4sin(a5Xn + a6) + a7sin(a8Yn + a9)

(Eq. 7G)

Yn+1 = a10 + a11Xn + a12Yn + a13sin(a14Xn + a15) + a16sin(a17Yn + a18)

It is not necessary to consider separately the cosine because the phase terms (a6 a9 a15 and a18) have the same effect since cos X is equal to sin(X + ¹/2). The third and fourth dimensions are determined in the same way as in Section 7.1.

The program modifications required to extend the computer search to such cases are shown in the listing PROG25. These cases will be coded with the backslash (\) which is ASCII 92.

PROG25 Changes Required in PROG24 to Search for Special Functions of the \ Type

1000 REM SPECIAL FUNCTION SEARCH (Sines and Cosines)

1090 ODE% = 5                   'System is special function \
3680 IF Q$ = "D" THEN D% = 1 + (D% MOD 10): T% = 1
6350 IF ODE% <> 5 THEN GOTO 6390
6360    M% = 18
6370    XNEW = A(1) + A(2) * X + A(3) * Y + A(4) * SIN(A(5) * X + A(6)) + A(7) * SIN(A(8) * Y + A(9))
6380    YNEW = A(10) + A(11) * X + A(12) * Y + A(13) * SIN(A(14) * X + A(15)) + A(16) * SIN(A(17) * Y + A(18))
6390 RETURN

Examples of attractors produced by PROG25 are shown in Figures 7-25 through 7-32.

\CDYVWGQ

Figure 7-25. Four-Dimensional Special Map \

\GHBBADG

Figure 7-26. Four-Dimensional Special Map \

\HPBAJVW

Figure 7-27. Four-Dimensional Special Map \

\JEUALUV

Figure 7-28. Four-Dimensional Special Map \

\RQYUANR

Figure 7-29. Four-Dimensional Special Map \

\TXBGOYY

Figure 7-30. Four-Dimensional Special Map \

\UMLPFJY

Figure 7-31. Four-Dimensional Special Map \

\WJPOONB

Figure 7-32. Four-Dimensional Special Map \

7.5 Webs and Wreaths

In this section we will consider a special type of map that involves sines and cosines. The solutions are chaotic but they are not attractors. The systems they describe are Hamiltonian. Such Hamiltonian systems obey the Liouville theorem which states that the phase-space volume occupied by a set of points will be conserved as the system evolves in time. Thus the orbit will eventually return arbitrarily close to an initial condition. Contrast this to dissipative systems in which the phase-space volume decreases in time eventually collapsing all initial conditions within the basin of attraction onto the attractor. In dissipative systems the basin of attraction is usually much larger than the attractor. The equations are as follows:

Xn+1 = 10a1 + [Xn + a2sin(a3Yn + a4)]cos a + Yn sin a

(Eq. 7H)

Yn+1 = 10a5 - [Xn + a2sin(a3Yn + a4)]sin a + Yn cos a

where a = 2¹ / (13 + 10a6). The third and fourth dimensions are determined in the same way as in Section 7.1.

The special form of Equations 7H not only guarantees that the solution is area-preserving but that it has circular symmetry. Furthermore there is an inherent periodicity that arises from the fact that a is 2¹ divided by an integer that ranges from 1 to 25. The periodicity is indicated by the last letter of the code (a6): A for period-1 B for period-2 and so forth. Because of the circular symmetry and infinite extent it is interesting to project these cases onto a sphere using the P command.

The program modifications required to extend the computer search to such cases are shown in the listing PROG26. These cases will be coded with the right bracket (]) which is ASCII 93.

PROG26 Changes Required in PROG25 to Search for Special Functions of the ] Type

1000 REM SPECIAL FUNCTION SEARCH (Webs and Wreaths)

1090 ODE% = 6                   'System is special function ]
1150 TWOPI = 6.28318530717959#  'A useful constant (2 pi)
3680 IF Q$ = "D" THEN D% = 1 + (D% MOD 11): T% = 1
6390 IF ODE% <> 6 THEN GOTO 6450
6400    M% = 6
6410    IF N < 2 THEN AL = TWOPI / (13 + 10 * A(6)): SINAL = SIN(AL): COSAL = COS(AL)
6420    DUM = X + A(2) * SIN(A(3) * Y + A(4))
6430    XNEW = 10 * A(1) + DUM * COSAL + Y * SINAL
6440    YNEW = 10 * A(5) - DUM * SINAL + Y * COSAL
6450 RETURN

As you watch the patterns develop you will see the orbit wander throughout the XY-plane along a network of channels. The network is infinite in extent and is called a stochastic web. The global wandering is evidence of minimal chaos and it causes the orbit eventually to leave the boundary of the computer screen. If you interrupt the calculation at some point the resulting structure resembles a wreath or a snowflake. The infinite structure is a tiling but its symmetry is slightly spoiled by the finite thickness of the web. This breaking of the symmetry eliminates the monotony and contributes to the aesthetic appeal of the patterns.

The slow wandering of the orbit throughout the web is an example of Arnol'd diffusion which is named after the Russian mathematician Vladimir Arnol'd. Normally we associate diffusion with a random process in which for example the molecules of a gas move slowly from one region to another by countless collisions with other molecules. The presence of diffusion in such simple deterministic systems has many practical consequences such as providing a means for heating a gas of electrically charged particles (a plasma) in a magnetic field using electromagnetic waves.

These stochastic webs contain circular chains of islands or beads on a necklace if you prefer a different analogy whose interior contains periodic orbits. Surrounding the islands is a stochastic sea in which the orbits are chaotic and connected to all other points in the sea. You will also note that the Lyapunov exponents are small. Since the orbits diffuse slowly in the sea nearby orbits remain close together for many iterations. For a similar reason the calculated fractal dimension is lower than it should be. Recall that the dimension calculation is based on the previous 500 iterates whose values tend to be nearly equal in this case.

Examples of stochastic webs produced by PROG26 are shown in Figures 7-33 through 7-40. These cases because of the slow diffusion of the orbit provide a good opportunity to exhibit the time variation with colors as shown in Plates 31 and 32. You may wish to try different values of NMAX% in line 1050 to control the rate at which the colors change. Web maps provide a perfect illustration of how chaos and determinism coexist. The underlying symmetry of the equations is evident in the figures but the orbit exhibits apparently random motion within the chaotic region.

]DBYHVD

Figure 7-33. Four-Dimensional Web Map

]DTDQCM

Figure 7-34. Four-Dimensional Web Map

]ICUEBE

Figure 7-35. Four-Dimensional Web Map

]KUTEEL

Figure 7-36. Four-Dimensional Web Map

]PFXOTL

Figure 7-37. Four-Dimensional Web Map

]PHXUEG

Figure 7-38. Four-Dimensional Web Map

]SRBOSW

Figure 7-39. Four-Dimensional Web Map

]XEUOII

Figure 7-40. Four-Dimensional Web Map

7.6 Swings and Springs

All the previous examples of maps and differential equations in this book share the property that the right-hand side of the equations is independent of the iteration number (N) or time (t). Such equations are called autonomous. For a given set of initial conditions they produce the same solution for whatever time or iteration number they are started.

Some important physical processes are most conveniently expressed by non-autonomous equations. An example is a driven (forced) damped linear harmonic oscillator which is described by the following equations:

X' = Y

(Eq. 7I)

Y' = -X - bY + A sin wt

In Equations 7I b is the damping constant (friction) A is the amplitude of the drive (forcing) function and w is the angular frequency (radians per second) of the drive. The friction force is assumed to be proportional and opposite to the velocity (-Y) although other forms give qualitatively similar results. This type of friction is called linear damping.

The usual trick for dealing with non-autonomous equations is to introduce an additional variable (say Z) and rewrite Equations 7I for example as

X' = Y

(Eq. 7J)

Y' = -X - bY + A sin Z

Z' = w

Equations 7J contain a nonlinearity (sin Z) but they do not have chaotic solutions. The solution (for positive b) is a limit cycle with frequency w. The limit cycle is largest when the damping is small (b much less than 1) and w is 1 corresponding to resonance.

To obtain interesting chaotic solutions we need additional nonlinear terms. We will restrict these nonlinearities to odd polynomials (X X3 X2Y and so forth) in the second equation to preserve the symmetry of the oscillation. A general form with odd polynomials up to order 3 is as follows:

X' = a1Y

(Eq. 7K)

Y' = a2X + a3X3 + a4X2Y + a5XY2 + a6Y + a7Y3 + a8sin Z

Z' = a9 + 1.3

If the product a1a2 is negative these equations represent the motion of a mass oscillating on a nonlinear spring or a pendulum swinging through a large angle (but not going over the top). If a3/a2 is positive the spring gets stiffer when stretched or compressed (hard spring). If a3/a2 is negative the spring gets weaker when stretched or compressed (soft spring). If a3/a2 is -1/6 the solution approximates a vigorously swinging pendulum. If the product a1a2 is positive and a3/a2 is negative the system models a buckled beam and is called the Duffing two-well oscillator. The final combination (a1a2 and a3/a2 both positive) is unstable and has an unbounded solution.

In Equations 7K we can exploit the fact that Z enters only through the term sin Z and thus it is periodic with period 2p. Whenever Z exceeds 2p we can subtract 2p from it without changing the result. This trick keeps Z bounded in the range 0 to 2p rather than letting it march off to infinity as it would otherwise do. The Z-coordinate then becomes the phase angle of the drive function. Note that the MOD function in most versions of BASIC works correctly only on integer variables and thus it should not be used for the above purpose. The +1.3 term in the Z-equation ensures that the phase angle always increases in time for -1.2 < a9 < 1.2. The fourth variable (W) is proportional to time as in the previous examples.

The program modifications required to extend the computer search to such cases are shown in the listing PROG27. These cases will be coded with the circumflex (^) which is ASCII 94.

PROG27 Changes Required in PROG26 to Search for Special Functions of the ^ Type

1000 REM SPECIAL FUNCTION SEARCH (Swings and Springs)

1090 ODE% = 7                   'System is special function ^
3050    IF ODE% = 1 OR ODE% = 7 THEN L = L / EPS
3680 IF Q$ = "D" THEN D% = 1 + (D% MOD 12): T% = 1
6450 IF ODE% <> 7 THEN GOTO 6500
6460    M% = 9
6470    XNEW = X + EPS * A(1) * Y
6480    YNEW = Y + EPS * (A(2) * X + A(3) * X * X * X + A(4) * X * X * Y + A(5) * X * Y * Y + A(6) * Y + A(7) * Y * Y * Y + A(8) * SIN(Z))
6490    ZNEW = Z + EPS * (A(9) + 1.3): IF ZNEW > TWOPI THEN ZNEW = ZNEW - TWOPI
6500 RETURN

Examples of attractors produced by PROG27 are shown in Figures 7-41 through 7-48. These cases are displayed as slices that show the orbit at 16 different drive phases. As you scan across them left to right and top to bottom you will see the stretching and folding that are characteristics of strange attractors and that account for their fractal micro-structure and for the sensitivity to initial conditions.

^GHXTOKL

Figure 7-41. Slices of Four-Dimensional Special Map ^

^VSFFNSL

Figure 7-42. Slices of Four-Dimensional Special Map ^

^WQEODHN

Figure 7-43. Slices of Four-Dimensional Special Map ^

^XXDJQPI

Figure 7-44. Slices of Four-Dimensional Special Map ^

^YTEODIM

Figure 7-45. Slices of Four-Dimensional Special Map ^

^YVFQFFL

Figure 7-46. Slices of Four-Dimensional Special Map ^

^YXBHQPE

Figure 7-47. Slices of Four-Dimensional Special Map ^

^YXEHASJ

Figure 7-48. Slices of Four-Dimensional Special Map ^

These cases provide an ideal opportunity to animate the third dimension (drive phase). If you have not fanned through the figures in the upper right-hand corner of the odd pages of the book do so now. Be sure to fan in the forward direction (low to high page numbers). These figures were produced using sixty-four phase slices of the attractor ^VYGJBPIJN with ten million iterations.

You will see several cycles of stretching and folding. Note that there are three diagonal bands that run from the lower left to the upper right of the attractor. The three bands are stretched and compressed into a single band while two additional bands enter first from the upper left and then from the lower right. Thus one of the three bands consists of three smaller bands one of which consists of three even smaller bands and so forth. Such an infinitely layered band is called a thick line or a Cantor one-manifold. Viewed in three dimensions the thick line would appear as a thick surface or a Cantor two-manifold. There is perhaps no clearer illustration anywhere in this book of the way strange attractors are formed and acquire their fractal micro-structure.

7.7 Roll Your Own

Perhaps this is a good place to leave you with this brief taste of the immense variety of nonlinear functions and equations nearly all of which admit strange attractors that can be identified and examined using the technique described in this book. The possibilities are limited only by your imagination and you can be assured that nearly every strange attractor that you discover will never have been seen before. There surely exist classes of objects yet to be discovered that are of mathematical and artistic interest.

If you decide to pursue such an exploration you might start with some of the other nonlinear functions and operators that are built into the BASIC language. Table 7.1 lists a number of interesting possibilities. They are divided into mathematical functions which should be independent of the machine or programming language machine functions which are dependent on the machine or language nonlinear operators which are supported by nearly all versions of BASIC and advanced functions which are supported by Visual BASIC for MS-DOS. You may use them in combinations to invent complicated forms that belong to you alone!

Table 7.1 Nonlinear Functions and Operators Supported by Most Versions of BASIC

CHAPTER 8

Epilogue

It would be an injustice to leave you with the impression that the main use of the ideas in this book is to make pretty pictures. This final chapter will describe some of the scientific applications of the method that was used to generate this large collection of strange attractors. It will also suggest some additional studies that you might wish to undertake as an extension of both the scientific and artistic aspects of the work described in this book.

8.1 How Common is Chaos?

For hundreds of years scientists in many disciplines have used equations like those in the previous chapters to describe nature. It is remarkable that almost no one recognized the chaotic solutions to those equations until the last few decades of the twentieth century. Now people are beginning to see chaos under every rock. It is reasonable to wonder whether chaos is the rule or the exception.

Suppose we had a system of equations of sufficient complexity and with sufficiently many coefficients that it could be used to model most natural processes. We could then attempt to quantify the occurrence of chaos in these equations and draw an inference about the occurrence of chaos in nature. The equations in the previous chapters especially those involving polynomials of high dimension and high order might approximate such a system.

As a simple but very unrealistic example suppose that all of nature could be modeled by the logistic equation (Equation 1C). This equation has a single parameter R that controls the character of the solution. If R is greater than 4 or less than -2 the solutions are unbounded which means that this equation cannot model a physical process under those conditions. Essentially all physical processes are bounded except perhaps the trajectory of a spacecraft launched with sufficient velocity to escape the galaxy. In the physically realistic range of R there is a band of chaos between about 3.5 and 4 as shown in Figure 1-2 and another identical band between -2 and about -1.5. Careful analysis shows that chaos occurs over about 13% of the range of R from -2 to 4.

Since the logistic equation is too simple a model for almost everything we should examine more complicated models. The Hénon map (Equations 3A) is a two-dimensional generalization of the logistic map. It has two control parameters which are normally taken as a = -1.4 and b = 0.3. With b = 0 the Hénon map reduces to the logistic map.

As with the logistic map the Hénon map has unbounded solutions chaotic solutions and bounded non-chaotic solutions depending on the values of a and b. The bounded non-chaotic solutions may be either fixed points or periodic limit cycles. Figure 8-1 shows a region of the ab-plane with the four classes of solutions indicated by different shades of gray. The bounded solutions constitute an island in the ab-plane. On the northwest shore of this island is a chaotic beach which occupies about 6% of the area of the island. The chaotic beach has many small embedded periodic ponds. The boundary between the chaotic and the periodic regions is itself a fractal.

b versus a plot

Figure 8-1. Regions of Solutions for the Hénon Map in the ab-Plane

The logistic map is chaotic over 13% of its bounded range and the Hénon map is chaotic over 6% of its bounded range. This result is counter-intuitive since it suggests that more complicated (2-D) systems are in some sense less chaotic than simpler (1-D) systems. Is this a general result or is it peculiar to these two maps? One way to decide is to examine a wider selection of equations such as the ones used to produce the attractors exhibited throughout this book.

In collecting attractors we have been discarding interesting information--the number of bounded non-chaotic solutions for each chaotic case that the program finds. Discarding data is offensive to scientists since experiments are often performed with great effort and at considerable expense. The annals of science are ripe with examples of important discoveries that could have been made sooner or by others if only the right data had been recorded and analyzed.

Table 8.1 shows the results from 30 000 chaotic cases (1000 for each of the 30 types) as identified by the program. This table required examining over 400 million cases of which about one million were bounded. Of all the bounded solutions 2.8% were chaotic according to the criterion described in Section 2.4. The polynomial maps all exhibit a similar occurrence of chaotic solutions. The same is true of ordinary differential equations (ODEs) but the percentage is smaller. The reason for this behavior is not understood.

Table 8.1 Summary of Data from 30 000 Chaotic Cases

Code D O Type Chaotic Average F Average L

A 1 2 Map 3.34% 0.81±0.15 0.53±0.42

B 1 3 Map 5.09% 0.80±0.14 0.50±0.40

C 1 4 Map 8.09% 0.82±0.12 0.52±0.20

D 1 5 Map 7.94% 0.80±0.14 0.51±0.21

E 2 2 Map 7.58% 1.20±0.32 0.27±0.16

F 2 3 Map 7.08% 1.19±0.33 0.27±0.15

G 2 4 Map 6.40% 1.16±0.32 0.27±0.15

H 2 5 Map 5.79% 1.19±0.30 0.28±0.16

I 3 2 Map 6.68% 1.50±0.40 0.16±0.10

J 3 3 Map 5.89% 1.45±0.39 0.15±0.09

K 3 4 Map 5.08% 1.45±0.41 0.15±0.09

L 3 5 Map 4.68% 1.43±0.39 0.14±0.09

M 4 2 Map 4.99% 1.64±0.47 0.10±0.06

N 4 3 Map 4.78% 1.59±0.46 0.09±0.06

O 4 4 Map 5.32% 1.61±0.45 0.09±0.06

P 4 5 Map 5.04% 1.62±0.47 0.10±0.06

Q 3 2 ODE 0.55% 1.28±0.41 0.21±0.33

R 3 3 ODE 1.33% 1.31±0.40 0.73±0.76

S 3 4 ODE 1.23% 1.35±0.40 1.05±0.98

T 3 5 ODE 1.63% 1.38±0.41 1.23±1.16

U 4 2 ODE 1.34% 1.43±0.43 0.16±0.23

V 4 3 ODE 1.84% 1.43±0.43 0.40±0.48

W 4 4 ODE 1.84% 1.46±0.45 0.54±0.62

X 4 5 ODE 1.96% 1.44±0.44 0.66±0.76

Y 4 Special 16.28% 1.37±0.56 0.26±0.26

Z 4 Special 23.19% 1.03±0.44 0.28±0.44

[ 4 Special 16.00% 0.63±0.65 0.42±0.23

\ 4 Special 1.61% 1.10±0.28 0.16±0.10

] 4 Special 19.80% 1.02±0.16 0.06±0.04

^ 4 Special 1.91% 1.80±0.49 0.39±1.03

These results should not be taken too seriously because the coefficients have been limited to the range -1.2 to 1.2 the ODEs have not been solved very accurately and there are many ambiguous cases. There is a tendency for the chaotic solutions to occur at large values of the coefficients where most of the solutions are unbounded. A more careful evaluation correcting these difficulties and including about 35 000 strange attractors but limited to fewer types shows that the probability that a bounded solution is chaotic for an iterated polynomial map of dimension D and order O is given approximately by

P = 0.349 D-1.69 O-0.28 (Eq. 8A)

Similarly the probability that a bounded solution is chaotic for a polynomial ODE of dimension D and order O is given approximately by

P = 0.0003 D2 O0.5 (Eq. 8B)

Maps appear to become less chaotic as they become more complicated (larger D and O) whereas ODEs become more chaotic.

To assess how common chaos is in nature we would need to address the more complicated and subjective issue of whether the equations we have examined are a representative sample of the equations that describe natural processes. Furthermore there is no reason to assume a priori that nature selects the coefficients of the equations uniformly over the bounded region of control space. It is possible that other constraints mitigate either against or in favor of chaotic behavior.

Another interesting question is how the fractal dimension and the Lyapunov exponent vary with the dimension and order of the system. Table 8.1 includes the average values of these quantities plus or minus (±) the standard deviation for each type of chaotic system. For polynomial maps and ODEs the fractal dimension varies approximately as the square root of the embedding dimension. For polynomial maps the Lyapunov exponent varies inversely with the embedding dimension. For polynomial ODEs the Lyapunov exponent increases with embedding dimension. The Lyapunov exponent appears to be independent of order for maps but there is a tendency for the Lyapunov exponent of ODEs to increase with order.

These results are summarized in Figures 8-2 and 8-3. Figure 8-2 shows the relative probability that a strange attractor from a polynomial map or ODE will have a fractal dimension F plotted versus F/D0.5. The curve is sharply peaked at a value of about 0.8. There are almost no attractors with fractal dimension greater than about 1.2D0.5. The Lorenz and Rössler Attractors (with fractal dimensions slightly above 2.0 in a three-dimensional space) are close to this maximum value. Figure 8-3 shows the relative probability that a strange attractor from a polynomial map will have a Lyapunov exponent L plotted versus LD. This curve shows a much broader peak at about 0.5. The reason for this behavior is not understood.

P versus F/ÃD

Figure 8-2. Probability Distribution of Fractal Dimension

P versus L*D

Figure 8-3. Probability Distribution of Lyapunov Exponent

The similarity of the fractal dimension for attractors produced by polynomials of the same dimensionality raises the concern that they might in some sense all be the same attractor viewed from different angles and distorted in various ways. There may be a simple mapping that converts one attractor into the others. In such a case a statistical analysis of the collection would be misleading and meaningless. However since the Lyapunov exponents are spread over a broad range it seems likely that the attractors are distinct. In any case they are visually very different and thus the technique has artistic if not scientific value.

It is interesting to ask whether the above results are peculiar to polynomials. Table 8.1 includes data for the special functions that were described in Chapter 7. Some of these cases tend to be more chaotic than the polynomials but the differences are not enormous. Thus is would appear insofar as nature can be represented by systems of equations of the type described in this book that chaos is not the most common behavior but neither is it particularly rare.

8.2 But is it Art?

A very different question is whether pictures generated by solving deterministic equations with a computer can legitimately qualify as art. Some people would say that if it was done by a computer without human intervention it cannot be art. On the other hand humans chose the equations built and programmed the computer decided how the solution would be displayed and selected from the large number of cases that the computer generated. In this view the computer is just another tool in the hands of the artist.

At the core of the issue is what we mean by art. There are at least two not necessarily mutually incompatible views. One is that art is the expression of ideas and emotions--a form of communication between the artist and the observer. The other emphasizes formal design in which the viewer admires the skill with which the artist manipulated the materials without reading any particular meaning into it.

Strange attractors qualify by either definition. They are expressions of ideas embodied in the equations whether it be the dynamics of population growth or a swinging pendulum. These ideas are often abstract and are most apparent to the trained mathematician or scientist but anyone can see in the patterns the surreal images of plants animals clouds and swirling fluids. The appearance of such images in the solutions of mathematical equations is probably more than coincidental.

Strange attractors also necessarily embody concepts of design. The interplay of determinism and unpredictability ensures that they are neither formless nor excessively repetitious. Furthermore the skills of an artisan (if not an artist) are required to translate the abstract equations into aesthetically desirable visual forms. These skills are different from (but not inferior to) those possessed by more conventional artists. Renaissance artists such as Leonardo da Vinci were often also scientists. We may now be entering a new Renaissance in which art and science are again being drawn together through the visual images produced by computers.

Some artists view art as a creative process whose primary goal is to provide the artist with a sense of satisfaction. The resulting work is merely an inevitable by-product. This view seems especially appropriate for the production of strange attractors where the programmer's satisfaction is derived from causing the computer to generate the patterns even if they are never seen by anyone else. Indeed the computer offers the ideal medium for such conceptual artists since there need be no material product whatsoever.

Note that visual art does not need to be beautiful to be good just as a play need not be humorous. It may be intellectually or emotionally satisfying or even disturbing. It should capture and hold the interest of the viewer however. The span of the viewer's attention is one measure of its quality. Art may mix the familiar and the unfamiliar to produce both comfort and dissonance. Furthermore beauty is at least partially in the eye of the beholder although recent research indicates that there are absolute universal measures of beauty that form very early in life and that may even be genetic.

8.3 Can Computers Critique Art?

The idea that a computer can make aesthetic judgments seems absurd and even offensive to many people. Yet the program developed in this book is already doing this to some degree. For every object (strange attractor) that it identifies it has discarded many dozens as being uninteresting (non-chaotic). Perhaps the computer could be programmed to be even more discriminating and to select those strange attractors that are likely to appeal to humans. To the extent that aesthetic judgment involves objective as well as subjective criteria such a proposition is not unreasonable.

A computer lacks emotion but it can be taught in much the same way that people can be taught. With the help of a human to point out which attractors are visually interesting the computer can correlate human opinion with various quantitative measures of the attractor. It can then test each new case and assign a probability that it would appeal to a human.

One of the reasons we have been calculating and saving the fractal dimension and Lyapunov exponent for each strange attractor is in anticipation of developing such criteria. You can think of the dimension as a measure of the strangeness of an attractor and the Lyapunov exponent as a measure of its chaoticity. These are just two of infinitely many independent quantities that we could use to describe each attractor. If we find encouragement from them it suggests that more could be done.

The first step is to search for a relation between the aesthetic quality and the fractal dimension or Lyapunov exponent. For this purpose 7500 strange attractors from two-dimensional quadratic maps were evaluated by the author and seven volunteers including two graduate art students a former art history major three physics graduate students and a former mathematics major. All evaluators were born and raised in the United States. The evaluations were done by choosing attractors randomly and displaying them sequentially on the computer screen without any indication of the quantities that characterize them. The volunteers were asked to evaluate each case on a scale of one to five according to its aesthetic appeal. It only took a few seconds for each evaluation.

Figure 8-4 displays a summary of all the evaluations as a function of fractal dimension (F) and Lyapunov exponent (L) using a gray scale in which the darker regions are the most highly rated. The cases examined by particular individuals show a similar trend. All evaluators tended to prefer attractors with dimensions between about 1.1 and 1.5 and Lyapunov exponents between zero and about 0.3. Some of the most interesting cases have Lyapunov exponents below about 0.1. You will find many such examples earlier in this book.

L versus F plot

Figure 8-4. Regions of Highest Aesthetic Quality in the FL-Plane

The dimension preference is not surprising since many natural objects have dimensions in this range. Nature is strange but usually not totally bizarre. Some of the attractors that are most universally liked resemble well-known and easily recognizable objects. You will find many such examples in this book.

The Lyapunov exponent preference is harder to understand but it suggests that strongly chaotic systems are too unpredictable to be appealing. For the 443 cases that were rated five (best) by the evaluators the average dimension was F = 1.30 ± 0.20 and the average Lyapunov exponent was L = 0.21 ± 0.13 bits per iterations where the errors represent plus or minus one standard deviation. About 28% of the cases evaluated fall within both error bars. Thus this simple criterion would allow the computer to discard nearly three-quarters of the cases that are least likely to be visually appealing. This technique works in practice and was used in some cases to help select attractors to display in this book.

8.4 What's Left to Do?

This book has described a new technique for generating strange attractors in unlimited numbers. A large collection of such objects offers many interesting possibilities to the artist and scientist alike. Some of these uses have been mentioned previously. Others may already have occurred to you. This section will leave you with a few additional suggestions of things you might want to explore on your own.

If your main interest is art you probably would like to produce attractors with improved spatial resolution and with more colors. You can experiment with printing on different types of paper or other media. The simple linear correspondence of X Y Z and W to position or color is not essential. Other mappings between the mathematical variables and the points displayed on the screen are possible. You have already seen how to project the attractors onto a sphere. You can project them onto other objects such as cylinders tori or even other strange attractors.

When you have generated an attractor that appeals to you it is natural to want to make small changes to make it even better. The coefficients in the equations are controls that you can adjust. Like knobs on your television set they allow you to tune the attractor to get just what you want. Similarly you will want more control over the colors which is possible with the PALETTE command in BASIC. You may also want to rotate the image to find the best angle from which to view it.

You can produce animated strange attractors with a video camera viewing a monitor connected to the camera or even more simply with a photo-diode connected to an oscilloscope whose screen illuminates the photo-diode. This video-feedback technique has much in common with iterated maps. Each illuminated dot on the screen is mapped back to a different location after a delay determined by the propagation of the electrical and optical signals. The main control parameters are the distance rotation focusing intensity color and hue. Some settings produce unbounded solutions--the screen goes solid black or solid white. Other settings produce a fixed-point solution with a stationary pattern. Under other conditions periodic behavior occurs. The most interesting situation is when the pattern constantly changes but is not periodic corresponding to a chaotic strange attractor. Sometimes you can perturb the system with a flash of light or by moving your hand across the field of view and cause the system to switch from one attractor to another.

Some of the most interesting examples of computer fractals come from plotting the basin boundaries of various attractors. The basin is the set of all initial conditions that are drawn to the attractor. Sometimes these boundaries are smooth; other times they are fractals. By coloring the points just outside the basin according to the number of iterations required for them to leave some (usually large) region surrounding the attractor beautiful escape-time fractal patterns can be produced.

All of our attractors have such basins and it's not hard to program the computer to display them but the calculations are very slow. As an example Figure 8-5 shows in black the basin for the Hénon map. It is relatively smooth and not particularly interesting. It resembles a thick version of the attractor but rotated by 90 degrees. Figure 8-6 shows the basin of another two-dimensional quadratic map called the Tinkerbell map. Its boundary has obvious fractal structure. In each case the basin boundary appears to touch the attractor suggesting that this surprising feature may be common.

EWM?MPMM

Figure 8-5. Basin of Attraction for the Hénon Map

EMVWMGCM

Figure 8-6. Basin of Attraction for the Tinkerbell Map

Notice that we have now plotted the Hénon map in three different spaces. Figure 3-1 is the usual plot in the space of the dynamical variables X and Y. Figure 8-1 is a plot in the space of the control parameters a and b. Figure 8-5 is a plot in the space of the initial conditions X0 and Y0. All of these plots are necessary to characterize the attractor completely.

The well-known and much-studied Mandelbrot set is the set of bounded solutions in the ab-plane of the mapping

Xn+1 = Xn2 - Yn2 + a

(Eq. 8C)

Yn+1 = 2XnYn + b

The variables X and Y are usually thought of as the real and imaginary parts of a complex number Z. Each point in the ab-plane has associated with it a Julia set which is the set of all initial conditions X0 and Y0 whose solutions are bounded. The Julia set is named after Gaston Julia a French mathematician who along with Pierre Fatou studied iterated maps in the complex plane at about the time of the first world war. Combinations of a and b at the boundary of the Mandelbrot set produce chaotic solutions but values inside the set lead to fixed points or limit cycles. For b = 0 and Y0 = 0 (the real axis) Equations 8C reduce to a simple one-dimensional quadratic map equivalent to the logistic map.

The Mandelbrot set has been described as the most complicated mathematical object ever seen. It may also be the ultimate computer virus in that it takes over not only the machine (because of the large computational requirements) but the mind of the programmer who often becomes addicted to the beauty and variety of the patterns that it produces.

Like the Mandelbrot set most of the attractors produced by the programs in this book have intricate structure near the basin boundaries. You can zoom in on these regions to produce patterns that rival those produced by the Mandelbrot set. Calculation times increase markedly as you zoom in ever more closely however.

Much more could be done to correlate the aesthetic appeal of the attractors with the various numerical quantities that characterize them. Besides the fractal dimension and Lyapunov exponent there is the dimension and order of the system that produced them. There are other measures of an attractor's dimension such as the capacity dimension the information dimension and the Lyapunov dimension. Related to the Lyapunov exponent is the entropy which is a measure of the disorder of the system. As with the dimension there are many ways to define the entropy. These quantities and others are described in many of the books on fractals in the bibliography. Maps and differential equations could also be compared.

You could test for differences between the aesthetic preferences of artists and scientists. Preliminary indications suggest that complexity might appeal more to artists than to scientists who tend to see beauty in simplicity. There may also be discernible cultural differences. You could see how the various display techniques alter one's preferences. For example the use of color seems to increase the tolerance for attractors of high fractal dimension.

There are many scientific studies that could be performed on a sufficiently large collection of strange attractors. The results of Table 8.1 which are merely suggestive and limited to particular types of equations could be refined with more cases and more examples of each case. For low dimensions maps are more chaotic than the equivalent ODEs. As the embedding dimension increases however maps become less chaotic while ODEs become more chaotic. Equations 8A and 8B suggest that for an embedding dimension of about six maps and ODEs will be equally chaotic. Will their roles reverse at higher dimension or will the chaotic fractions asymptotically approach a universal value of about two percent?

The same type of questions could be asked about bounded non-chaotic solutions (fixed points limit cycles and tori). It is especially interesting to see how common 3-tori are in light of Peixoto's theorem (see Section 6.6) but their production probably requires embedding dimensions of about 10 according to Figure 8-2 which may or may not apply to tori. Almost nothing is known about such issues.

We have considered the fraction of the bounded hyper-dimensional control space over which chaos occurs. It is also interesting to examine the shape location and dimension of this chaotic region as we did for the two-parameter Hénon map in Figure 8-1. It is likely that this region is a fractal. How does its fractal dimension depend on the embedding dimension and other characteristics of the system of equations? It appears that the dimension of the chaotic region is about half the dimension of the control space for the cases in this book.

The scaling of average fractal dimension and Lyapunov exponent with embedding dimension is intriguing and should be studied with more cases better statistics and more accurate calculations of the fractal dimension. You could examine statistically how the various measures of dimension compare and how these dimensions are related to the spectrum of Lyapunov exponents.

You could look for relations between the geometrical properties of attractors and their respective power spectra. The technique could be used to explore and to quantify the various routes by which a stable solution becomes chaotic. Many such routes have been identified such as the period-doubling exhibited by the logistic equation but there are probably others yet to be discovered. Your role resembles that of a biologist confronted with a large variety of species trying to classify quantify and study their similarities differences and patterns of behavior.

Another interesting study is to search for previously unknown simple examples of chaos. Variations of the logistic equation are the simplest chaotic polynomial maps. The Lorenz and Rössler attractors are often cited as the simplest examples of chaotic polynomial ODEs. The Lorenz equations (Eq. 6D) have seven terms and two quadratic nonlinearities and the Rössler equations (Eq. 6E) have seven terms and one quadratic nonlinearity.

There are at least five different systems of chaotic three-dimensional quadratic ODEs with five terms and two nonlinearities and five systems with six terms and one nonlinearity. It will be left as a challenge for you to find them. There does not seem to be any chaotic system of polynomial ODEs with as few as four terms. To whet your appetite here's a simple system resembling the Lorenz attractor that has two fewer terms and all its coefficients equal to one:

X' = YZ

(Eq. 8D)

Y' = X - Y

Z' = 1 - XY

You can examine this case using the code QM7NM3NM3LM4NM2LM6. Here's another case with five terms and all unity coefficients that's volume-preserving and thus does not have an attractor but its solution is either a two-torus or chaotic depending on the initial conditions:

X' = Y

(Eq. 8E)

Y' = -X + YZ

Z' = 1 - Y2

It can be produced with the code QM5NM5LM5NM2NM5LM3.

8.5 What Good is it?

It is rare that a mathematical concept captivates the interest not only of scientists in diverse fields but of the general public. Chaos has done this and it has been heralded by some as the next great revolution in science. It has ushered in a new field of experimental mathematics sometimes called recreational mathematics by the more traditional and often cynical older breed of mathematicians. The use of computers to produce exotic visual patterns has made difficult mathematical ideas accessible to those without extensive formal training.

Yet it is fair to ask what it has done other than to make pretty pictures. One response is to claim that the same things were probably said about the discovery of the atomic nucleus--or electricity--or fire. We must have faith that intellectual advances will eventually yield useful applications.

At the deepest level an understanding of chaos alters our view of the world. Having seen the complexity that can arise from simple equations we have reason to hope that simple equations may suffice to describe much of the complexity of the world. Difficult and long-standing problems such as fluid turbulence may eventually be understood using the ideas of chaos. Turbulence is difficult because it involves both temporal and spatial chaos and because its dimension is high. Chaos may also provide tools for making better predictions of complicated and apparently random systems such as the weather the stock market earthquakes epidemics and population growth. It may also have applications in cryptography for devising codes that are difficult to break and for breaking existing codes.

Since chaotic systems exhibit extreme sensitivity to initial conditions you might conclude that prediction is hopeless but this is not the case. Prediction is hopeless for a random system or for an extremely complicated deterministic system. As the physicist Neils Bohr remarked "Prediction is difficult especially of the future." However if the system is simple and chaotic then the determinism can be exploited to improve short-term predictions.

Furthermore if the equations produce a strange attractor as most chaotic systems do we know that the solution will lie somewhere on the attractor. Remember that an attractor occupies a negligible volume of the space in which it is embedded. Although we can't predict where on the attractor the system will be at any particular time in the distant future we can exclude the vast number of possible states that lie off the attractor. In meteorological terms this might lead to eliminating the possibility of certain weather conditions with near absolute certainty.

When we see a system in nature that behaves erratically we are now led to wonder whether it is an example of chaos. Is determinism hidden in seemingly random data? This problem is opposite to the one addressed in this book. Here we have started with simple equations and produced complicated patterns. Nature presents us with complicated patterns that may or may not come from simple equations.

Often we are given only a single fluctuating quantity sampled at discrete times--a so-called time series. The average daily temperature in New York and the daily closing Dow Jones Industrial Average are two examples of a time series. Sometimes a complicated-looking time series is simply a sum of several sine waves of different unrelated frequencies. In such a case accurate predictions are possible using linear methods. A good example is the tides whose behavior is not simply periodic but which nevertheless can be predicted with considerable accuracy. More often the time series has no such simple representation. In such a case we would like to determine whether the behavior is random or chaotic. Chaotic systems often produce strange attractors.

We usually don't have data for all the variables that describe the system nor do we even know how many there are and so we don't know the dimension of the space in which an attractor might be embedded. Furthermore the data record is likely to be corrupted by random noise and measurement errors. The number of data points may be small and the system may not have reached a steady-state. Finally we usually don't have control over the system and so we cannot directly test its sensitivity to initial conditions.

Such a situation sounds hopeless but progress has recently been made in detecting chaos in natural systems. For example one can plot each data point versus its predecessor as we did with the one-dimensional maps in Chapter 2. In some cases this procedure is enough to reveal the determinism. More often it is necessary to plot each data point versus several of its predecessors in a high-dimensional space. If the system is described by a strange attractor the result will be a distorted version of the attractor but with the same fractal dimension. Similarly the Lyapunov exponent can be estimated by selecting nearby points in this space and determining how rapidly they diverge from one another.

The fractal dimension is important because the number of variables and equations is at least as large as the next higher integer since the attractor has to be embedded in a space with dimension higher than its fractal dimension. These equations are not unique however. Extracting equations from the data is a difficult if not impossible task but one whose rewards justify the effort.

Dynamical systems are everywhere. Your body contains several. Your heart is a dynamical system whose solution normally approaches a limit cycle but which can become chaotic when in fibrillation or a fixed point upon death. Recent research suggests that even a healthy heart beats chaotically and that a nearly periodic pattern sometimes precedes cardiac arrest.

Other oscillations occur in your lungs brain and muscles. Electrocardiograms and electroencephalograms are time-series records used by doctors to deduce information about the dynamics of the corresponding organ. Some chaos in the brain may be necessary for creativity in that it prevents us from repeatedly approaching the same problems in the same way. The fractal dimension of electroencephalograms has been observed to increase when a subject is engaged in mental activity. In psychology manic depression resembles a limit cycle and certain types of erratic behavior might be strange attractors.

Evidence for low-dimensional strange attractors has been found in systems as diverse as the weather and climate the business cycle childhood epidemics sunspots plasma fluctuations and even in computer models of the arms race. Some of these results should be viewed skeptically because there are many ways to be misled. One needs the order of 10D data points where D is the embedding dimension although the exact requirements are unknown. If the interval between data points is too small a fictitiously low dimension can result. Certain types of colored noise (also called fractional Brownian motion) have a degree of determinism and produce time-series records with an apparently low dimension. Many tests have been devised such as comparing the results with those from a surrogate data set obtained by randomly shuffling the numbers in the time series.

Chaos has a deep philosophical significance. If determinism implied perfect predictability there would be no room for free will. We would be just small cogs in a large machine over which we have no control. It's easy to feel insignificant in a vast and complicated universe dominated by forces too powerful to resist.

However chaos illustrates that determinism does not imply predictability. What is bad news for those who want to predict the weather or the stock market is good news for those who want to control them. Recall the hypothetical butterfly flapping its wings in Brazil which sets off tornadoes in Texas. If the world is really governed by deterministic chaos we are drastically changing the future with everything we do. We need never feel unimportant or insignificant. Who would have thought that a concept from mathematics could give meaning and purpose to our lives?

APPENDIX A

Annotated Bibliography

Books and articles on chaos and fractals have proliferated enormously in recent years. A selection of the most useful and readable of these is listed below along with comments on the content and level of each.

E. A. Abbott: Flatland: A Romance of Many Dimensions 1884 (Reprint Barnes & Noble New York 1983). This entertaining classic by a Victorian schoolmaster and minister explains the fourth and higher dimensions by considering the impact of a visit by a three-dimensional creature (A Sphere) to a world inhabited by two-dimensional creatures (A Square and others).

F. D. Abraham: A Visual Introduction to Dynamical Systems Theory for Psychology (Aerial Press Santa Cruz CA 1990). This book shows how attractors in dynamical systems have application to a field as unlikely as psychology. It contains numerous sketches of attractors and an explanation of the technical terms used to describe them.

R. H. Abraham and C. D. Shaw: Dynamics--The Geometry of Behavior (Aerial Press Santa Cruz CA 1982). This four-part series attempts to explain the fundamentals of dynamical behavior without equations using cartoon-like drawings with extended captions.

M. Barnsley: Fractals Everywhere (Academic Press San Diego 1988). This classic text by an expert mathematician describes the mathematics underlying fractals and provides a good source of new fractal types.

M. F. Barnsley R. L. Devaney B. B. Mandelbrot H. -O. Peitgen D. Saupe and R. F. Voss: The Science of Fractal Images (Springer-Verlag New York 1988). This readable collection of articles describes the mathematics that underlies the computer generation of fractals.

J. Briggs and F. Peat: The Turbulent Mirror: An Illustrated Guide to Chaos Theory and the Science of Wholeness (Harper and Row New York 1989). This unusual book combines a simple mathematical description of chaos with loosely related philosophical and psychological ramblings.

D. Burger: Sphereland (Barnes and Noble New York 1983). This entertaining book in the style of the one by Abbott picks up where Flatland stops in describing the properties of high-dimensional spaces.

A. A. Chernikov R. Z. Sagdeev and G. M. Zaslavsky: "Chaos: How Regular Can it Be?" (Physics Today 27 November 1988). This review article by a group of active Russian nonlinear dynamicists provides a good background for understanding stochastic webs.

J. P. Crutchfield J. D. Farmer N. H. Packard and R. S. Shaw: "Chaos" (Scientific American 255 46 December 1986). This easily readable article presents the fundamentals of chaos theory in a compact form.

R. L. Devaney: An Introduction to Chaotic Dynamical Systems (Addison-Wesley Redwood City CA 1989). This advanced undergraduate text summarizes the mathematics that underlies chaotic systems of equations.

R. L. Devaney: Chaos Fractals and Dynamics: Computer Experiments in Mathematics (Addison-Wesley Menlo Park CA 1990). This book introduces chaos fractals and nonlinear dynamics with numerous beautiful images using computer exercises in BASIC and simple mathematics.

R. L. Devaney: A First Course in Chaotic Dynamical Systems: Theory and Experiment (Addison Wesley New York 1992). This book at the undergraduate college level provides an excellent introduction to chaos and fractals.

A. K. Dewdney: "Probing the Strange Attractors of Chaos" (Scientific American 257 108 July 1987). This typical Scientific American presentation provides a readable though abbreviated introduction to the mathematics underlying strange attractors.

K. Falconer: Fractal Geometry: Mathematical Foundations and Applications (Wiley New York 1990). This somewhat technical book explains the mathematics underlying fractals and describes the various methods of defining and calculating their dimension.

J. Feder: Fractals (Plenum Press New York 1988). This introductory technical book describes fractals using geometrical ideas and explains time-series analysis and other related topics from a physical viewpoint.

M. J. Feigenbaum: "Qualitative Universality for a Class of Nonlinear Transformations" (Journal of Statistical Physics 19 25 1978). This highly technical but historically important paper lays the foundation for many of the principles of chaos by reference to the logistic equation and similar one-dimensional nonlinear maps.

M. Field and M. Golubitsky: Symmetry in Chaos: A Search for Pattern in Mathematics Art and Nature (Oxford University Press Oxford 1992). This spectacularly illustrated book includes computer-generated images of flowers graphic logos motifs and other artistic mathematical patterns.

J. Gleick: Chaos: Making a New Science (Viking Penguin New York 1987). This best-selling historical and non-technical account is the starting point for anyone who wants to understand why chaos has excited the imagination of the scientist and nonscientist alike.

P. Grassberger and I. Procaccia: "Characterization of Strange Attractors" (Physical Review Letters 50 346 1983). This somewhat mathematical paper was the first to propose the correlation dimension as a convenient method for quantifying strange attractors.

B. L. Hao: Chaos II (World Scientific Singapore 1990). This book reprints many introductory and influential papers on chaos and includes a bibliography of 117 books and 2244 technical papers.

M. Hénon: "A Two-Dimensional Mapping with a Strange Attractor" (Communications in Mathematical Physics 50 69 1976). This somewhat technical article describes one of the first thoroughly studied examples of a chaotic two-dimensional quadratic map.

D. R. Hofstadter: Godel Escher Bach: An Eternal Golden Braid (Vintage New York 1980). This highly recommended book which deals among other things with self-similarity and recursion in music art and science is largely non-mathematical but deeply philosophical.

D. R. Hofstadter: "Strange Attractors: Mathematical Patterns Delicately Poised between Order and Chaos" (Scientific American 245 22 November 1981). This article provides an introduction to the mathematics of strange attractors and their significance.

D. R. Hofstadter: Metamagical Themas (Basic Books New York 1985). This recreational mathematics book contains many interesting topics including a chapter on mathematical chaos and strange attractors.

E.A. Jackson: Perspectives of Nonlinear Dynamics (Cambridge University Press Cambridge 1991). This two-volume graduate level text covers all the essentials of chaos and nonlinear dynamics with many nice drawings.

D. E. Knuth: Seminumerical Algorithms 2nd ed. vol. 2 of The Art of Computer Programming (Addison-Wesley Reading MA 1981) Chap. 3. This serious mathematical book is the classic reference for describing among other things the methods by which computers are used to generate pseudo-random numbers and the pitfalls inherent in their misuse.

S. Krasner ed.: The Ubiquity of Chaos (American Association for the Advancement of Science Washington D. C. 1990). This collection of nineteen scientific papers illustrates the wide range of fields in which chaotic processes have been discovered and studied.

S. Levy: Artificial Life: The Quest for a New Creation (Pantheon Books New York 1992). This book takes up where Gleick left off and covers much of the recent history of chaos and its developers.

T. -Y. Li and J. A. Yorke: "Period Three Implies Chaos" (American Mathematical Monthly 82 985 1975). This historical paper contains the first published reference to the term chaos.

E. N. Lorenz: "Deterministic Nonperiodic Flow" (Journal of the Atmospheric Sciences 20 130 1963). This historically important paper was slow to be appreciated but is now regarded as the first modern example of a strange attractor resulting from the solution of a simple set of ordinary differential equations originally proposed to model atmospheric convection.

B. B. Mandelbrot: The Fractal Geometry of Nature (W. H. Freeman San Francisco 1982). This extended essay by the father of fractals was the seminal work that brought to the attention of the non-specialist the ubiquity of fractals in nature.

R. M. May: "Simple Mathematical Models with Very Complicated Dynamics" (Nature 261 459 1976). This thought-provoking and mathematically straightforward paper initiated the widespread interest in the logistic equation and other one-dimensional maps as models for natural processes.

M. McGuire: An Eye for Fractals ( Addison-Wesley Reading MA 1991). This graphic and photographic essay by a physicist and amateur photographer contains beautiful black and white photographs of natural fractals in the style of Ansel Adams with a simple mathematical description of fractals.

F. C. Moon: Chaotic Vibrations (Wiley-Interscience New York 1987). This advanced undergraduate text emphasizes the applications of chaos theory to real-world engineering problems.

H. -O. Peitgen and P. H. Richter: The Beauty of Fractals: Images of Complex Dynamical Systems (Springer-Verlag Berlin 1986). This beautiful colorful exhibit of computer art emphasizes the Mandelbrot and Julia sets.

H. -O. Peitgen H. Jurgens and D. Saupe: Fractals for the Classroom (Springer-Verlag New York 1992). This two-book set explains the mathematical basis of fractals at a relatively elementary level and contains many BASIC program listings for the production of fractals.

C. A. Pickover: Computers Pattern Chaos and Beauty: Graphics from an Unseen World (St. Martin's Press New York 1990). This "how-to" book by the master of computer graphic art and visualization is filled with original ideas for the computer generation of fractals and other artistic patterns.

C. A. Pickover: Computers and the Imagination: Visual Adventures Beyond the Edge (St. Martin's Press New York 1991). This extension of Pickover's earlier book by the same publisher takes up where the other stopped and provides additional spectacular examples of computer art and philosophical insight.

C. A. Pickover: Mazes for the Mind: Computers and the Unexpected (St. Martin's Press New York 1992). This third book in the series includes puzzles games and mazes appealing to computer enthusiasts artists and puzzle-solvers along with incisive commentary and additional dazzling computer images.

E. Porter and J. Gleick: Nature's Chaos (Viking Penguin New York 1990). This book of art combines photographs of natural fractals by Porter with a simple almost poetic explanation of chaos and fractals by Gleick.

J. Pritchard: The Chaos Cookbook: A Practical Programming Guide (Butterworth-Heinemann Oxford 1992). This practical and elementary tutorial includes programs in BASIC and Pascal that ease the reader into the mathematics of chaos and fractals.

O. E. Rössler: "An Equation for Continuous Chaos" (Physics Letters 57A 397 1976). This short classic paper by a non-practicing medical doctor includes stereoscopic views of the Lorenz and Rössler attractors.

R. Rucker: The Fourth Dimension (Houghton-Mifflin New York 1984). This book is the place to go if you want to understand the philosophical geometrical and physical meaning of space-time.

D. Ruelle: "Strange Attractors" (Mathematical Intelligencer 2 126 1980). This mathematical article includes some of the history of the discovery and understanding of strange attractors and related chaotic phenomena.

M. Schroeder: Fractals Chaos and Power Laws: Minutes from an Infinite Paradise (W. H. Freeman New York 1991). This slightly mathematical but highly readable book is packed with examples of temporal and spatial chaos in enormously diverse contexts and a wealth of puns.

H. G. Schuster: Deterministic Chaos (Springer-Verlag New York 1984). This work is aimed at the more mathematically inclined reader who wants to understand Chaos and related topics in greater detail and it includes a good mathematical description of the Lyapunov exponent.

R. Shaw: The Dripping Faucet as a Model Chaotic System (Aerial Press Santa Cruz CA 1984). This short but detailed book describes in non-technical language how the ideas of chaos and strange attractors can be used to understand a phenomenon as simple as a dripping faucet.

C. T. Sparrow: The Lorenz Equations Bifurcations Chaos and Strange Attractors (Springer-Verlag New York 1982). This somewhat technical book demonstrates the depth to which a single example of a strange attractor can be studied.

J. C. Sprott: "Simple Programs Create 3-D Images" (Computers in Physics 6 132 1992). This article by the author describes in detail how computers can be programmed to produce anaglyphic images of 3-D objects.

J. C. Sprott: "How Common is Chaos?" (Physics Letters A 173 21 1993). This abbreviated technical paper quantifying the occurrence of chaos in polynomial maps and ODEs was the inspiration for this book.

J. C. Sprott and G. Rowlands: Chaos Demonstrations (Physics Academic Software North Carolina State University Raleigh NC 27695-8202). This IBM PC computer program by the author and a colleague provides a simple way to learn about chaos fractals and related phenomena. It comes with an 84-page User's manual and 3-D glasses.

R. T. Stevens: Fractal Programming in C (M&T Books Redwood City CA 1989). This book which is also available in a TurboPascal version provides programs and detailed descriptions for producing most of the standard fractal forms.

R. T. Stevens: Advanced Fractal Programming in C (M&T Books Redwood City CA 1990). This book picks up where the previous one left off with emphasis on Mandelbrot and Julia sets but with discussion of other fractal types such as L-systems and iterated function systems.

I. Stewart: Does God Play Dice?: The Mathematics of Chaos (Blackwell Oxford 1989). This charming light-hearted but serious book provides a good introduction to chaos using simple mathematics.

I. Stewart and M. Golubitsky: Fearful Symmetry: Is God a Geometer? (Blackwell Publishers Oxford 1992). This sequel to the earlier book by Stewart deals with symmetry in nature art and science and provides computer programs for producing symmetrical patterns of considerable beauty.

J. Theiler: "Estimating Fractal Dimension" (Journal of the Optical Society of America 7A 1055 1990). This slightly technical paper contains an excellent review of the various ways of calculating fractal dimensions.

J. M. T. Thompson and H. B. Stewart: Nonlinear Dynamics and Chaos (Wiley New York 1986). This advanced undergraduate text aimed at physical science students explains the mathematical basis for chaos.

A. A. Tsonis: Chaos: From Theory to Applications (Plenum Press New York 1992). This advanced undergraduate text includes recent developments in the applications of chaos theory to real-world problems such as improved methods of forecasting and distinguishing chaos from noise.

J. R. Weeks: The Shape of Space: How to Visualize Surfaces and Three-Dimensional Manifolds (Marcel Dekker Incorporated New York 1985). This book explains in simple language and with clear illustrations the elements of topology that are fundamental to a deep understanding of strange attractors and other geometrical objects.

T. Wegner and M. Peterson: Fractal Creations (Waite Group Press Mill Valley CA 1991). This book is really a user's manual for the incredible freeware program FRACTINT which is constantly updated by a dedicated group of volunteer fractal programming enthusiasts.

A. Wolf J. B. Swift H. L. Swinney and J. A. Vastano: "Determining Lyapunov Exponents from a Time Series" (Physica 16D 285 1985). This technical but surprisingly readable article contains an excellent description of the practical considerations that go into the calculation of Lyapunov exponents.

S. Y. Zhang: Bibliography on Chaos (World Scientific Singapore 1991). This bibliography of over 7000 references is the most extensive compilation available and it serves to underscore the popularity of the subject.

APPENDIX B

BASIC Program Listing

This appendix contains the complete BASIC program that you should have developed if you followed the exercises in this book along with a few additions. It should run without modification on DOS-based IBM personal computers or compatibles under Microsoft BASICA GW-BASIC QBASIC QuickBASIC or VisualBASIC for MS-DOS; Borland International TurboBASIC; and Spectra Publishing PowerBASIC. A disk containing the source program and a version of the program complied with PowerBASIC (SA.EXE) and ready to run is included with the book. This is a relatively no-frills program in that it lacks extensive error trapping fancy menus and mouse support but it is fully functional and relatively robust.

The additions to the program are as follows:

1. The program contains a version number (2.0) and a copyright notice. Your purchase of this book and the accompanying disk entitles you to personal use of the program. It is not legal for you to make a copy of the program for someone else to place it in the public domain or to incorporate it in whole or in part into programs that are distributed to others. The idea of programming a computer to search automatically for strange attractors based on calculation of the Lyapunov exponent is believed to be original and proper scientific etiquette requires that you acknowledge the author in any further dissemination of work based on this technique.

2. The program includes a somewhat inelegant but effective test for the graphics capability of the computer on which it is used. It causes the program to run automatically in the highest graphics mode that is supported by the hardware and by the BASIC version under which it is compiled or run. The program prints a message and stops if the computer does not have a graphics monitor. Colors are adjusted for CGA MODE 1 and text is properly formatted for screens with 40 columns of text.

3. The program allows you to change the number of iterations that are plotted while in the search mode using the N key. Values of a thousand (10^3) to a billion (10^9) are allowed. Note that this value excludes the thousand iterations that are always performed to allow the initial transient to decay.

4. In addition to the planar and spherical projections the P command allows you to project the attractors onto a cylinder with a horizontal or vertical axis or a torus. The toroidal projection is shown looking along the major axis with the doughnut hole at the center of the screen and of negligible size.

5. A C command has been added to allow you to clear the screen and restart the calculation with the current values of the variables. This feature allows you to remove the transient in cases where the orbit requires more than a thousand iterations to reach the attractor or to reduce the density of points on the screen which is sometimes useful for example with the 3-D anaglyphic displays.

6. The program allows you to press V to save a record of up to 16 000 consecutive iterates of X Y Z or W in a disk file that can be analyzed in more detail by other programs. To conserve disk space each new attractor overwrites the data from the previous case. The data files can be read by the companion program Chaos Data Analyzer which allows the data to be displayed in many ways including phase-space plots return maps and Poincaré movies; calculates probability distributions power spectra Lyapunov exponents correlation functions and capacity and correlation dimensions; and makes predictions based on a novel technique involving singular value decomposition. Chaos Data Analyzer is available from The Academic Software Library Box 8202 North Carolina State University Raleigh NC 27695-8202 Tel. (800) 955-TASL or (919) 515-7447.

If you have been working systematically through the programs in this book you will find useful the following list of program lines that require changes to produce the final program PROG28.BAS:

1000 1070 1090 1140 1310 1330 1340 1360 2270-2290 2500 3060 3360 3400-3420 3630 3670 3740 3750 3780 4000 4240 4250 4280 4380 4390 4430-4450 4570-4590 5650-5710 5840 6600-7070

PROG28.BAS Complete BASIC Program for Producing all the Examples in this Book and Endless Variations

1000 REM STRANGE ATTRACTOR PROGRAM  BASIC Ver 2.0 (c) 1993 by J. C. Sprott

1010 DEFDBL A-Z                 'Use double precision
1020 DIM XS(499)
YS(499)
ZS(499)
WS(499)
A(504)
V(99)
XY(4)
XN(4)
COLR%(15)
1030 SM% = 12                   'Assume VGA graphics
1040 PREV% = 5                  'Plot versus fifth previous iterate
1050 NMAX = 11000               'Maximum number of iterations
1060 OMAX% = 5                  'Maximum order of polynomial
1070 D% = 2                     'Dimension of system
1080 EPS = .1                   'Step size for ODE
1090 ODE% = 0                   'System is map
1100 SND% = 0                   'Turn sound off
1110 PJT% = 0                   'Projection is planar
1120 TRD% = 1                   'Display third dimension as shadow
1130 FTH% = 2                   'Display fourth dimension as colors
1140 SAV% = 0                   'Don't save any data
1150 TWOPI = 6.28318530717959#  'A useful constant (2 pi)
1160 RANDOMIZE TIMER            'Reseed random number generator
1170 GOSUB 4200                 'Display menu screen
1180 IF Q$ = "X" THEN GOTO 1250 'Exit immediately on command
1190 GOSUB 1300                 'Initialize
1200 GOSUB 1500                 'Set parameters
1210 GOSUB 1700                 'Iterate equations
1220 GOSUB 2100                 'Display results
1230 GOSUB 2400                 'Test results
1240 ON T% GOTO 1190
1200
1210
1250 CLS
1260 END
1300 REM Initialize
1310 ON ERROR GOTO 6600         'Find legal graphics mode
1320 SCREEN SM%                 'Set graphics mode
1330 ON ERROR GOTO 0            'Resume default error trapping
1340 DEF SEG = 64: WID% = PEEK(74)  'Number of text columns
1350 WINDOW (-.1
-.1)-(1.1
1.1)
1360 CLS : LOCATE 13
WID% / 2 - 6: PRINT "Searching..."
1370 GOSUB 5600                 'Set colors
1380 IF QM% <> 2 THEN GOTO 1420
1390    NE = 0: CLOSE
1400    OPEN "SA.DIC" FOR APPEND AS #1: CLOSE
1410    OPEN "SA.DIC" FOR INPUT AS #1
1420 RETURN
1500 REM Set parameters
1510 X = .05                    'Initial condition
1520 Y = .05
1530 Z = .05
1540 W = .05
1550 XE = X + .000001: YE = Y: ZE = Z: WE = W
1560 GOSUB 2600                 'Get coefficients
1570 T% = 3
1580 P% = 0: LSUM = 0: N = 0: NL = 0: N1 = 0: N2 = 0
1590 XMIN = 1000000!: XMAX = -XMIN: YMIN = XMIN: YMAX = XMAX
1600 ZMIN = XMIN: ZMAX = XMAX
1610 WMIN = XMIN: WMAX = XMAX
1620 TWOD% = 2 ^ D%
1630 RETURN
1700 REM Iterate equations
1710 IF ODE% > 1 THEN GOSUB 6200: GOTO 2020     'Special function
1720 M% = 1: XY(1) = X: XY(2) = Y: XY(3) = Z: XY(4) = W
1730 FOR I% = 1 TO D%
1740 XN(I%) = A(M%)
1750 M% = M% + 1
1760 FOR I1% = 1 TO D%
1770 XN(I%) = XN(I%) + A(M%) * XY(I1%)
1780 M% = M% + 1
1790 FOR I2% = I1% TO D%
1800 XN(I%) = XN(I%) + A(M%) * XY(I1%) * XY(I2%)
1810 M% = M% + 1
1820 IF O% = 2 THEN GOTO 1970
1830 FOR I3% = I2% TO D%
1840 XN(I%) = XN(I%) + A(M%) * XY(I1%) * XY(I2%) * XY(I3%)
1850 M% = M% + 1
1860 IF O% = 3 THEN GOTO 1960
1870 FOR I4% = I3% TO D%
1880 XN(I%) = XN(I%) + A(M%) * XY(I1%) * XY(I2%) * XY(I3%) * XY(I4%)
1890 M% = M% + 1
1900 IF O% = 4 THEN GOTO 1950
1910 FOR I5% = I4% TO D%
1920 XN(I%) = XN(I%) + A(M%) * XY(I1%) * XY(I2%) * XY(I3%) * XY(I4%) * XY(I5%)
1930 M% = M% + 1
1940 NEXT I5%
1950 NEXT I4%
1960 NEXT I3%
1970 NEXT I2%
1980 NEXT I1%
1990 IF ODE% = 1 THEN XN(I%) = XY(I%) + EPS * XN(I%)
2000 NEXT I%
2010 M% = M% - 1: XNEW = XN(1): YNEW = XN(2): ZNEW = XN(3): WNEW = XN(4)
2020 N = N + 1
2030 RETURN
2100 REM Display results
2110 IF N < 100 OR N > 1000 THEN GOTO 2200
2120    IF X < XMIN THEN XMIN = X
2130    IF X > XMAX THEN XMAX = X
2140    IF Y < YMIN THEN YMIN = Y
2150    IF Y > YMAX THEN YMAX = Y
2160    IF Z < ZMIN THEN ZMIN = Z
2170    IF Z > ZMAX THEN ZMAX = Z
2180    IF W < WMIN THEN WMIN = W
2190    IF W > WMAX THEN WMAX = W
2200 IF N = 1000 THEN GOSUB 3100        'Resize the screen
2210 XS(P%) = X: YS(P%) = Y: ZS(P%) = Z: WS(P%) = W
2220 P% = (P% + 1) MOD 500
2230 I% = (P% + 500 - PREV%) MOD 500
2240 IF D% = 1 THEN XP = XS(I%): YP = XNEW ELSE XP = X: YP = Y
2250 IF N < 1000 OR XP <= XL OR XP >= XH OR YP <= YL OR YP >= YH THEN GOTO 2320
2260    IF PJT% = 1 THEN GOSUB 4100     'Project onto a sphere
2270    IF PJT% = 2 THEN GOSUB 6700     'Project onto a horizontal cylinder
2280    IF PJT% = 3 THEN GOSUB 6800     'Project onto a vertical cylinder
2290    IF PJT% = 4 THEN GOSUB 6900     'Project onto a torus
2300    GOSUB 5000                      'Plot point on screen
2310    IF SND% = 1 THEN GOSUB 3500     'Produce sound
2320 RETURN
2400 REM Test results
2410 IF ABS(XNEW) + ABS(YNEW) + ABS(ZNEW) + ABS(WNEW) > 1000000! THEN T% = 2
2420 IF QM% = 2 THEN GOTO 2490  'Speed up evaluation mode
2430 GOSUB 2900                 'Calculate Lyapunov exponent
2440 GOSUB 3900                 'Calculate fractal dimension
2450 IF QM% > 0 THEN GOTO 2490  'Skip tests when not in search mode
2460 IF N >= NMAX THEN T% = 2: GOSUB 4900   'Strange attractor found
2470 IF ABS(XNEW - X) + ABS(YNEW - Y) + ABS(ZNEW - Z) + ABS(WNEW - W) < .000001 THEN T% = 2
2480 IF N > 100 AND L < .005 THEN T% = 2    'Limit cycle
2490 Q$ = INKEY$: IF LEN(Q$) THEN GOSUB 3600        'Respond to user command
2500 IF SAV% > 0 THEN IF N > 1000 AND N < 17001 THEN GOSUB 7000 'Save data
2510 X = XNEW                   'Update value of X
2520 Y = YNEW
2530 Z = ZNEW
2540 W = WNEW
2550 RETURN
2600 REM Get coefficients
2610 IF QM% <> 2 THEN GOTO 2640 'Not in evaluate mode
2620    IF EOF(1) THEN QM% = 0: GOSUB 6000: GOTO 2640
2630    IF EOF(1) = 0 THEN LINE INPUT #1
CODE$: GOSUB 4700: GOSUB 5600
2640 IF QM% > 0 THEN GOTO 2730  'Not in search mode
2650    O% = 2 + INT((OMAX% - 1) * RND)
2660    CODE$ = CHR$(59 + 4 * D% + O% + 8 * ODE%)
2670    IF ODE% > 1 THEN CODE$ = CHR$(87 + ODE%)
2680    GOSUB 4700              'Get value of M%
2690    FOR I% = 1 TO M%        'Construct CODE$
2700        GOSUB 2800          'Shuffle random numbers
2710        CODE$ = CODE$ + CHR$(65 + INT(25 * RAN))
2720    NEXT I%
2730 FOR I% = 1 TO M%           'Convert CODE$ to coefficient values
2740    A(I%) = (ASC(MID$(CODE$
I% + 1
1)) - 77) / 10
2750 NEXT I%
2760 RETURN
2800 REM Shuffle random numbers
2810 IF V(0) = 0 THEN FOR J% = 0 TO 99: V(J%) = RND: NEXT J%
2820 J% = INT(100 * RAN)
2830 RAN = V(J%)
2840 V(J%) = RND
2850 RETURN
2900 REM Calculate Lyapunov exponent
2910 XSAVE = XNEW: YSAVE = YNEW: ZSAVE = ZNEW: WSAVE = WNEW
2920 X = XE: Y = YE: Z = ZE: W = WE: N = N - 1
2930 GOSUB 1700                 'Reiterate equations
2940 DLX = XNEW - XSAVE: DLY = YNEW - YSAVE
2950 DLZ = ZNEW - ZSAVE: DLW = WNEW - WSAVE
2960 DL2 = DLX * DLX + DLY * DLY + DLZ * DLZ + DLW * DLW
2970 IF CSNG(DL2) <= 0 THEN GOTO 3070       'Don't divide by zero
2980    DF = 1000000000000# * DL2
2990    RS = 1 / SQR(DF)
3000    XE = XSAVE + RS * (XNEW - XSAVE): YE = YSAVE + RS * (YNEW - YSAVE)
3010    ZE = ZSAVE + RS * (ZNEW - ZSAVE): WE = WSAVE + RS * (WNEW - WSAVE)
3020    XNEW = XSAVE: YNEW = YSAVE: ZNEW = ZSAVE: WNEW = WSAVE
3030    LSUM = LSUM + LOG(DF): NL = NL + 1
3040    L = .721347 * LSUM / NL
3050    IF ODE% = 1 OR ODE% = 7 THEN L = L / EPS
3060    IF N > 1000 AND N MOD 10 = 0 THEN LOCATE 1
WID% - 4: PRINT USING "##.##"; L;
3070 RETURN
3100 REM Resize the screen
3110 IF D% = 1 THEN YMIN = XMIN: YMAX = XMAX
3120 IF XMAX - XMIN < .000001 THEN XMIN = XMIN - .0000005: XMAX = XMAX + .0000005
3130 IF YMAX - YMIN < .000001 THEN YMIN = YMIN - .0000005: YMAX = YMAX + .0000005
3140 IF ZMAX - ZMIN < .000001 THEN ZMIN = ZMIN - .0000005: ZMAX = ZMAX + .0000005
3150 IF WMAX - WMIN < .000001 THEN WMIN = WMIN - .0000005: WMAX = WMAX + .0000005
3160 MX = .1 * (XMAX - XMIN): MY = .1 * (YMAX - YMIN)
3170 XL = XMIN - MX: XH = XMAX + MX: YL = YMIN - MY: YH = YMAX + 1.5 * MY
3180 WINDOW (XL
YL)-(XH
YH): CLS
3190 YH = YH - .5 * MY
3200 XA = (XL + XH) / 2: YA = (YL + YH) / 2
3210 IF D% < 3 THEN GOTO 3310
3220    ZA = (ZMAX + ZMIN) / 2
3230    IF TRD% = 1 THEN LINE (XL
YL)-(XH
YH)
COLR%(1)
BF: GOSUB 5400
3240    IF TRD% = 4 THEN LINE (XL
YL)-(XH
YH)
WH%
BF
3250    IF TRD% = 5 THEN LINE (XA
YL)-(XA
YH)
3260    IF TRD% <> 6 THEN GOTO 3310
3270        FOR I% = 1 TO 3
3280            XP = XL + I% * (XH - XL) / 4: LINE (XP
YL)-(XP
YH)
3290            YP = YL + I% * (YH - YL) / 4: LINE (XL
YP)-(XH
YP)
3300        NEXT I%
3310 IF PJT% <> 1 THEN LINE (XL
YL)-(XH
YH)

B
3320 IF PJT% = 1 AND TRD% < 5 THEN CIRCLE (XA
YA)
.36 * (XH - XL)
3330 TT = 3.1416 / (XMAX - XMIN): PT = 3.1416 / (YMAX - YMIN)
3340 IF QM% <> 2 THEN GOTO 3400         'Not in evaluate mode
3350    LOCATE 1
1: PRINT "<Space Bar>: Discard       <Enter>: Save";
3360    IF WID% < 80 THEN GOTO 3390
3370    LOCATE 1
49: PRINT "<Esc>: Exit";
3380    LOCATE 1
69: PRINT CINT((LOF(1) - 128 * LOC(1)) / 1024); "K left";
3390    GOTO 3430
3400 LOCATE 1
1: IF LEN(CODE$) < WID% - 18 THEN PRINT CODE$
3410 IF LEN(CODE$) >= WID% - 18 THEN PRINT LEFT$(CODE$
WID% - 23) + "..."
3420 LOCATE 1
WID% - 17: PRINT "F =": LOCATE 1
WID% - 7: PRINT "L ="
3430 TIA = .05               'Tangent of illumination angle
3440 XZ = -TIA * (XMAX - XMIN) / (ZMAX - ZMIN)
3450 YZ = TIA * (YMAX - YMIN) / (ZMAX - ZMIN)
3460 RETURN
3500 REM Produce sound
3510 FREQ% = 220 * 2 ^ (CINT(36 * (XNEW - XL) / (XH - XL)) / 12)
3520 DUR = 1
3530 IF D% > 1 THEN DUR = 2 ^ INT(.5 * (YH - YL) / (YNEW - 9 * YL / 8 + YH / 8))
3540 SOUND FREQ%
DUR: IF PLAY(0) THEN PLAY "MF"
3550 RETURN
3600 REM Respond to user command
3610 IF ASC(Q$) > 96 THEN Q$ = CHR$(ASC(Q$) - 32)   'Convert to upper case
3620 IF QM% = 2 THEN GOSUB 5800                 'Process evaluation command
3630 IF INSTR("ACDEHINPRSVX"
Q$) = 0 THEN GOSUB 4200   'Display menu screen
3640 IF Q$ = "A" THEN T% = 1: QM% = 0
3650 IF ODE% > 1 THEN D% = ODE% + 5
3660 IF ODE% = 1 THEN D% = D% + 2
3670 IF Q$ = "C" THEN IF N > 999 THEN N = 999
3680 IF Q$ = "D" THEN D% = 1 + (D% MOD 12): T% = 1
3690 IF D% > 6 THEN ODE% = D% - 5: D% = 4: GOTO 3710
3700 IF D% > 4 THEN ODE% = 1: D% = D% - 2 ELSE ODE% = 0
3710 IF Q$ = "E" THEN T% = 1: QM% = 2
3720 IF Q$ = "H" THEN FTH% = (FTH% + 1) MOD 3: T% = 3: IF N > 999 THEN N = 999: GOSUB 5600
3730 IF Q$ = "I" THEN IF T% <> 1 THEN SCREEN 0: WIDTH 80: COLOR 15
1: CLS : LINE INPUT "Code? "; CODE$: IF CODE$ = "" THEN Q$ = " ": CLS :  ELSE T% = 1: QM% = 1: GOSUB 4700
3740 IF Q$ = "N" THEN NMAX = 10 * (NMAX - 1000) + 1000: IF NMAX > 10 ^ 10 THEN NMAX = 2000
3750 IF Q$ = "P" THEN PJT% = (PJT% + 1) MOD 5: T% = 3: IF N > 999 THEN N = 999
3760 IF Q$ = "R" THEN TRD% = (TRD% + 1) MOD 7: T% = 3: IF N > 999 THEN N = 999: GOSUB 5600
3770 IF Q$ = "S" THEN SND% = (SND% + 1) MOD 2: T% = 3
3780 IF Q$ = "V" THEN SAV% = (SAV% + 1) MOD 5: FAV$ = CHR$(87 + SAV% MOD 4): T% = 3: IF N > 999 THEN N = 999
3790 IF Q$ = "X" THEN T% = 0
3800 RETURN
3900 REM Calculate fractal dimension
3910 IF N < 1000 THEN GOTO 4010         'Wait for transient to settle
3920 IF N = 1000 THEN D2MAX = (XMAX - XMIN) ^ 2 + (YMAX - YMIN) ^ 2 + (ZMAX - ZMIN) ^ 2 + (WMAX - WMIN) ^ 2
3930 J% = (P% + 1 + INT(480 * RND)) MOD 500
3940 DX = XNEW - XS(J%): DY = YNEW - YS(J%): DZ = ZNEW - ZS(J%): DW = WNEW - WS(J%)
3950 D2 = DX * DX + DY * DY + DZ * DZ + DW * DW
3960 IF D2 < .001 * TWOD% * D2MAX THEN N2 = N2 + 1
3970 IF D2 > .00001 * TWOD% * D2MAX THEN GOTO 4010
3980    N1 = N1 + 1
3990    F = .434294 * LOG(N2 / (N1 - .5))
4000    LOCATE 1
WID% - 14: PRINT USING "##.##"; F;
4010 RETURN
4100 REM Project onto a sphere
4110 TH = TT * (XMAX - XP)
4120 PH = PT * (YMAX - YP)
4130 XP = XA + .36 * (XH - XL) * COS(TH) * SIN(PH)
4140 YP = YA + .5 * (YH - YL) * COS(PH)
4150 RETURN
4200 REM Display menu screen
4210 SCREEN 0: WIDTH 80: COLOR 15
1: CLS
4220 WHILE Q$ = "" OR INSTR("AEIX"
Q$) = 0
4230    LOCATE 1
27: PRINT "STRANGE ATTRACTOR PROGRAM"
4240    PRINT TAB(27); "IBM PC BASIC Version 2.0"
4250    PRINT TAB(27); "(c) 1993 by J. C. Sprott"
4260    PRINT : PRINT
4270    PRINT TAB(27); "A: Search for attractors"
4280    PRINT TAB(27); "C: Clear screen and restart"
4290    IF ODE% > 1 THEN PRINT TAB(27); "D: System is 4-D special map "; CHR$(87 + ODE%); " ": GOTO 4320
4300    PRINT TAB(27); "D: System is"; STR$(D%); "-D polynomial ";
4310    IF ODE% = 1 THEN PRINT "ODE" ELSE PRINT "map"
4320    PRINT TAB(27); "E: Evaluate attractors"
4330    PRINT TAB(27); "H: Fourth dimension is ";
4340        IF FTH% = 0 THEN PRINT "projection"
4350        IF FTH% = 1 THEN PRINT "bands     "
4360        IF FTH% = 2 THEN PRINT "colors    "
4370    PRINT TAB(27); "I: Input code from keyboard"
4380    PRINT TAB(27); "N: Number of iterations is 10^";
4390    PRINT USING "#"; CINT(LOG(NMAX - 1000) / LOG(10))
4400    PRINT TAB(27); "P: Projection is ";
4410        IF PJT% = 0 THEN PRINT "planar   "
4420        IF PJT% = 1 THEN PRINT "spherical"
4430        IF PJT% = 2 THEN PRINT "horiz cyl"
4440        IF PJT% = 3 THEN PRINT "vert cyl "
4450        IF PJT% = 4 THEN PRINT "toroidal "
4460    PRINT TAB(27); "R: Third dimension is ";
4470        IF TRD% = 0 THEN PRINT "projection"
4480        IF TRD% = 1 THEN PRINT "shadow    "
4490        IF TRD% = 2 THEN PRINT "bands     "
4500        IF TRD% = 3 THEN PRINT "colors    "
4510        IF TRD% = 4 THEN PRINT "anaglyph  "
4520        IF TRD% = 5 THEN PRINT "stereogram"
4530        IF TRD% = 6 THEN PRINT "slices    "
4540    PRINT TAB(27); "S: Sound is ";
4550        IF SND% = 0 THEN PRINT "off"
4560        IF SND% = 1 THEN PRINT "on "
4570    PRINT TAB(27); "V: ";
4580        IF SAV% = 0 THEN PRINT "No data will be saved       "
4590        IF SAV% > 0 THEN PRINT FAV$; " will be saved in "; FAV$; "DATA.DAT"
4600    PRINT TAB(27); "X: Exit program"
4610    Q$ = INKEY$
4620    IF Q$ <> "" THEN GOSUB 3600     'Respond to user command
4630 WEND
4640 RETURN
4700 REM Get dimension and order
4710 D% = 1 + INT((ASC(LEFT$(CODE$
1)) - 65) / 4)
4720 IF D% > 6 THEN ODE% = ASC(LEFT$(CODE$
1)) - 87: D% = 4: GOSUB 6200: GOTO 4770
4730 IF D% > 4 THEN D% = D% - 2: ODE% = 1 ELSE ODE% = 0
4740 O% = 2 + (ASC(LEFT$(CODE$
1)) - 65) MOD 4
4750 M% = 1: FOR I% = 1 TO D%: M% = M% * (O% + I%): NEXT I%
4760 IF D% > 2 THEN FOR I% = 3 TO D%: M% = M% / (I% - 1): NEXT I%
4770 IF LEN(CODE$) = M% + 1 OR QM% <> 1 THEN GOTO 4810
4780    BEEP        'Illegal code warning
4790    WHILE LEN(CODE$) < M% + 1: CODE$ = CODE$ + "M": WEND
4800    IF LEN(CODE$) > M% + 1 THEN CODE$ = LEFT$(CODE$
M% + 1)
4810 RETURN
4900 REM Save attractor to disk file SA.DIC
4910 OPEN "SA.DIC" FOR APPEND AS #1
4920 PRINT #1
CODE$; : PRINT #1
USING "##.##"; F; L
4930 CLOSE #1
4940 RETURN
5000 REM Plot point on screen
5010 C4% = WH%
5020 IF D% < 4 THEN GOTO 5050
5030    IF FTH% = 1 THEN IF INT(30 * (W - WMIN) / (WMAX - WMIN)) MOD 2 THEN GOTO 5330
5040    IF FTH% = 2 THEN C4% = 1 + INT(NC% * (W - WMIN) / (WMAX - WMIN) + NC%) MOD NC%
5050 IF D% < 3 THEN PSET (XP
YP): GOTO 5330    'Skip 3-D stuff
5060 IF TRD% = 0 THEN PSET (XP
YP)
C4%
5070 IF TRD% <> 1 THEN GOTO 5130
5080    IF D% > 3 AND FTH% = 2 THEN PSET (XP
YP)
C4%: GOTO 5110
5090    C% = POINT(XP
YP)
5100    IF C% = COLR%(2) THEN PSET (XP
YP)
COLR%(3) ELSE IF C% <> COLR%(3) THEN PSET (XP
YP)
COLR%(2)
5110    XP = XP - XZ * (Z - ZMIN): YP = YP - YZ * (Z - ZMIN)
5120    IF POINT(XP
YP) = COLR%(1) THEN PSET (XP
YP)
0
5130 IF TRD% <> 2 THEN GOTO 5160
5140    IF D% > 3 AND FTH% = 2 AND (INT(15 * (Z - ZMIN) / (ZMAX - ZMIN) + 2) MOD 2) = 1 THEN PSET (XP
YP)
C4%
5150    IF D% < 4 OR FTH% <> 2 THEN C% = COLR%(INT(60 * (Z - ZMIN) / (ZMAX - ZMIN) + 4) MOD 4): PSET (XP
YP)
C%
5160 IF TRD% = 3 THEN PSET (XP
YP)
COLR%(INT(NC% * (Z - ZMIN) / (ZMAX - ZMIN) + NC%) MOD NC%)
5170 IF TRD% <> 4 THEN GOTO 5240
5180    XRT = XP + XZ * (Z - ZA): C% = POINT(XRT
YP)
5190    IF C% = WH% THEN PSET (XRT
YP)
RD%
5200    IF C% = CY% THEN PSET (XRT
YP)
BK%
5210    XLT = XP - XZ * (Z - ZA): C% = POINT(XLT
YP)
5220    IF C% = WH% THEN PSET (XLT
YP)
CY%
5230    IF C% = RD% THEN PSET (XLT
YP)
BK%
5240 IF TRD% <> 5 THEN GOTO 5280
5250    HSF = 2                 'Horizontal scale factor
5260    XRT = XA + (XP + XZ * (Z - ZA) - XL) / HSF: PSET (XRT
YP)
C4%
5270    XLT = XA + (XP - XZ * (Z - ZA) - XH) / HSF: PSET (XLT
YP)
C4%
5280 IF TRD% <> 6 THEN GOTO 5330
5290    DZ = (15 * (Z - ZMIN) / (ZMAX - ZMIN) + .5) / 16
5300    XP = (XP - XL + (INT(16 * DZ) MOD 4) * (XH - XL)) / 4 + XL
5310    YP = (YP - YL + (3 - INT(4 * DZ) MOD 4) * (YH - YL)) / 4 + YL
5320    PSET (XP
YP)
C4%
5330 RETURN
5400 REM Plot background grid
5410 FOR I% = 0 TO 15           'Draw 15 vertical grid lines
5420    XP = XMIN + I% * (XMAX - XMIN) / 15
5430    LINE (XP
YMIN)-(XP
YMAX)
0
5440 NEXT I%
5450 FOR I% = 0 TO 10           'Draw 10 horizontal grid lines
5460    YP = YMIN + I% * (YMAX - YMIN) / 10
5470    LINE (XMIN
YP)-(XMAX
YP)
0
5480 NEXT I%
5490 RETURN
5600 REM Set colors
5610 NC% = 15                   'Number of colors
5620 COLR%(0) = 0: COLR%(1) = 8: COLR%(2) = 7: COLR%(3) = 15
5630 IF TRD% = 3 OR (D% > 3 AND FTH% = 2 AND TRD% <> 1) THEN FOR I% = 0 TO NC%: COLR%(I%) = I% + 1: NEXT I%
5640 WH% = 15: BK% = 8: RD% = 12: CY% = 11
5650 IF SM% > 2 THEN GOTO 5720  'Not in CGA mode
5660    WID% = 80: IF D% < 3 THEN SCREEN 2: GOTO 5720
5670    IF (TRD% = 0 OR TRD% > 4) AND (D% = 3 OR FTH% <> 2) THEN SCREEN 2: GOTO 5720
5680    WID% = 40: SCREEN 1
5690    COLR%(0) = 0: COLR%(1) = 2: COLR%(2) = 1: COLR%(3) = 3
5700    WH% = 3: BK% = 0: RD% = 2: CY% = 1
5710    FOR I% = 4 TO NC%: COLR%(I%) = COLR%(I% MOD 4 + 1): NEXT I%
5720 RETURN
5800 REM Process evaluation command
5810 IF Q$ = " " THEN T% = 2: NE = NE + 1: CLS
5820 IF Q$ = CHR$(13) THEN T% = 2: NE = NE + 1: CLS : GOSUB 5900
5830 IF Q$ = CHR$(27) THEN CLS : GOSUB 6000: Q$ = " ": QM% = 0: GOTO 5850
5840 IF Q$ <> CHR$(27) AND INSTR("CHNPRVS"
Q$) = 0 THEN Q$ = ""
5850 RETURN
5900 REM Save favorite attractors to disk file FAVORITE.DIC
5910 OPEN "FAVORITE.DIC" FOR APPEND AS #2
5920 PRINT #2
CODE$
5930 CLOSE #2
5940 RETURN
6000 REM Update SA.DIC file
6010 LOCATE 11
9: PRINT "Evaluation complete"
6020 LOCATE 12
8: PRINT NE; "cases evaluated"
6030 OPEN "SATEMP.DIC" FOR OUTPUT AS #2
6040 IF QM% = 2 THEN PRINT #2
CODE$
6050 WHILE NOT EOF(1): LINE INPUT #1
CODE$: PRINT #2
CODE$: WEND
6060 CLOSE
6070 KILL "SA.DIC"
6080 NAME "SATEMP.DIC" AS "SA.DIC"
6090 RETURN
6200 REM Special function definitions
6210 ZNEW = X * X + Y * Y       'Default 3rd and 4th dimension
6220 WNEW = (N - 100) / 900: IF N > 1000 THEN WNEW = (N - 1000) / (NMAX - 1000)
6230 IF ODE% <> 2 THEN GOTO 6270
6240    M% = 10
6250    XNEW = A(1) + A(2) * X + A(3) * Y + A(4) * ABS(X) + A(5) * ABS(Y)
6260    YNEW = A(6) + A(7) * X + A(8) * Y + A(9) * ABS(X) + A(10) * ABS(Y)
6270 IF ODE% <> 3 THEN GOTO 6310
6280    M% = 14
6290    XNEW = A(1) + A(2) * X + A(3) * Y + (CINT(A(4) * X) AND CINT(A(5) * Y)) + (CINT(A(6) * X) OR CINT(A(7) * Y))
6300    YNEW = A(8) + A(9) * X + A(10) * Y + (CINT(A(11) * X) AND CINT(A(12) * Y)) + (CINT(A(13) * X) OR CINT(A(14) * Y))
6310 IF ODE% <> 4 THEN GOTO 6350
6320    M% = 14
6330    XNEW = A(1) + A(2) * X + A(3) * Y + A(4) * ABS(X) ^ A(5) + A(6) * ABS(Y) ^ A(7)
6340    YNEW = A(8) + A(9) * X + A(10) * Y + A(11) * ABS(X) ^ A(12) + A(13) * ABS(Y) ^ A(14)
6350 IF ODE% <> 5 THEN GOTO 6390
6360    M% = 18
6370    XNEW = A(1) + A(2) * X + A(3) * Y + A(4) * SIN(A(5) * X + A(6)) + A(7) * SIN(A(8) * Y + A(9))
6380    YNEW = A(10) + A(11) * X + A(12) * Y + A(13) * SIN(A(14) * X + A(15)) + A(16) * SIN(A(17) * Y + A(18))
6390 IF ODE% <> 6 THEN GOTO 6450
6400    M% = 6
6410    IF N < 2 THEN AL = TWOPI / (13 + 10 * A(6)): SINAL = SIN(AL): COSAL = COS(AL)
6420    DUM = X + A(2) * SIN(A(3) * Y + A(4))
6430    XNEW = 10 * A(1) + DUM * COSAL + Y * SINAL
6440    YNEW = 10 * A(5) - DUM * SINAL + Y * COSAL
6450 IF ODE% <> 7 THEN GOTO 6500
6460    M% = 9
6470    XNEW = X + EPS * A(1) * Y
6480    YNEW = Y + EPS * (A(2) * X + A(3) * X * X * X + A(4) * X * X * Y + A(5) * X * Y * Y + A(6) * Y + A(7) * Y * Y * Y + A(8) * SIN(Z))
6490    ZNEW = Z + EPS * (A(9) + 1.3): IF ZNEW > TWOPI THEN ZNEW = ZNEW - TWOPI
6500 RETURN
6600 REM Find legal graphics mode
6610 SM% = SM% - 1
6620 IF SM% = 0 THEN PRINT "This program requires a graphics monitor": STOP
6630 RESUME
6700 REM Project onto a horizontal cylinder
6710 PH = PT * (YMAX - YP)
6720 YP = YA + .5 * (YH - YL) * COS(PH)
6730 RETURN
6800 REM Project onto a vertical cylinder
6810 TH = TT * (XMAX - XP)
6820 XP = XA + .5 * (XH - XL) * COS(TH)
6830 RETURN
6900 REM Project onto a torus (unity aspect ratio)
6910 TH = TT * (XMAX - XP)
6920 PH = 2 * PT * (YMAX - YP)
6930 XP = XA + .18 * (XH - XL) * (1 + COS(TH)) * SIN(PH)
6940 YP = YA + .25 * (YH - YL) * (1 + COS(TH)) * COS(PH)
6950 RETURN
7000 REM Save data
7010 IF N = 1001 THEN CLOSE #3: OPEN FAV$ + "DATA.DAT" FOR OUTPUT AS #3
7020 IF SAV% = 1 THEN DUM = XNEW
7030 IF SAV% = 2 THEN DUM = YNEW
7040 IF SAV% = 3 THEN DUM = ZNEW
7050 IF SAV% = 4 THEN DUM = WNEW
7060 PRINT #3
CSNG(DUM)
7070 RETURN

APPENDIX C

Other Computers and BASIC Versions

Here are some considerations for using the programs with non-IBM-compatible computers and with different dialects of BASIC.

BASICA and GW-BASIC

These old versions of BASIC are mostly compatible with the program listings in this book. They do not support VGA graphics and the older versions don't even support EGA. Thus you may have to change SM% = 12 in line 1030 to a lower number. The listing PROG28.BAS automatically selects an appropriate graphics mode.

These versions of BASIC do not support strings longer than 255 characters. The easiest way to circumvent this problem is to limit the four-dimensional searches to cubic polynomials. Adding the following line to the program after line 2680 will accomplish this:

2685 IF M% > 253 THEN GOTO 2650

TurboBASIC and PowerBASIC

The program listings in this book are compatible with TurboBASIC and PowerBASIC except for a quirk with the CIRCLE command with VGA graphics that requires the following change in line 3320:

3320 IF PJT% = 1 AND TRD% < 5 THEN IF SM% < 11 THEN CIRCLE (XA YA) .36 * (XH - XL) ELSE CIRCLE (XA YA) .5 * (YH - YL)

VisualBASIC for MS-DOS

The programs as listed will compile and run directly under VisualBASIC for MS-DOS. This version of BASIC makes it easy for you to add pull-down menus dialog boxes and mouse support to give the user interface a more modern look and feel.

VisualBASIC for Windows

One way to convert the programs to run as Microsoft Windows applications is to use the utility TRNSLATE.EXE supplied with VisualBASIC for MS-DOS to translate the programs into VisualBASIC for Windows. There are many program differences that will have to be resolved however.

A VisualBASIC for Windows version of the listing PROG06 but without the keyboard strobe and sound capabilities is given in the listing PROG06VB.BAS. VisualBASIC for Windows 2.0 does not support the SOUND statement or INKEY$ function although these capabilities and many others are available through internal Windows drivers. You can use this listing as a starting point for converting the other programs in this book for use with Microsoft Windows. The accompanying disk contains a compiled version of this program in the file SAWIN.EXE. To run this program the VBRUN200.DLL run-time dynamic link library that comes with VisualBASIC version 2.00 must be in a directory in your search path.

PROG06VB.BAS VisualBASIC for Windows Version of PROG06

VERSION 2.00

Begin Form PROG06VB 
Caption         =   "Strange Attractors"
Height          =   4620
Left            =   828
LinkTopic       =   "Form1"
ScaleHeight     =   4200
ScaleWidth      =   6420
Top             =   1128
Width           =   6516
End
DefDbl A-Z
Sub Form_Activate ()
1000 Rem TWO-D MAP SEARCH  VisualBASIC Ver 1.0 (c) 1993 by J. C. Sprott
1020 ReDim XS(499)
A(504)
V(99)
1040 PREV% = 5                 'Plot versus fifth previous iterate
1050 NMAX = 11000              'Maximum number of iterations
1060 OMAX% = 2                 'Maximum order of polynomial
1070 D% = 2                    'Dimension of system
1160 Randomize Timer           'Reseed random number generator
1190 GoSub 1300                'Initialize
1200 GoSub 1500                'Set parameters
1210 GoSub 1700                'Iterate equations
1220 GoSub 2100                'Display results
1230 GoSub 2400                'Test results
1240 On T% GoTo 1190
1200
1210
1250 Cls
1260 End
1300 Rem Initialize
1320 Cls: Msg$ = "Searching..."
1350 CurrentX = (ScaleWidth - TextWidth(Msg$)) / 2
1360 CurrentY = (ScaleHeight - TextHeight(Msg$)) / 2
1370 Print Msg$
1420 Return
1500 Rem Set parameters
1510 X = .05                   'Initial condition
1520 Y = .05
1550 XE = X + .000001: YE = Y
1560 GoSub 2600                'Get coefficients
1570 T% = 3
1580 P% = 0: LSUM = 0:  N = 0:  NL = 0
1590 XMIN = 1000000!: XMAX = -XMIN:  YMIN = XMIN:  YMAX = XMAX
1630 Return
1700 Rem Iterate equations
1720 XNEW = A(1) + X * (A(2) + A(3) * X + A(4) * Y)
1730 XNEW = XNEW + Y * (A(5) + A(6) * Y)
1830 YNEW = A(7) + X * (A(8) + A(9) * X + A(10) * Y)
1930 YNEW = YNEW + Y * (A(11) + A(12) * Y)
2020 N = N + 1
2030 Return
2100 Rem Display results
2110 If N < 100 Or N > 1000 Then GoTo 2200
2120      If X < XMIN Then XMIN = X
2130      If X > XMAX Then XMAX = X
2140      If Y < YMIN Then YMIN = Y
2150      If Y > YMAX Then YMAX = Y
2200 If N = 1000 Then GoSub 3100       'Resize the screen
2210 XS(P%) = X
2220 P% = (P% + 1) Mod 500
2230 I% = (P% + 500 - PREV%) Mod 500
2240 If D% = 1 Then XP = XS(I%): YP = XNEW Else XP = X:  YP = Y
2250 If N < 1000 Or XP <= XL Or XP >= XH Or YP <= YL Or YP >= YH Then GoTo 2320
2300      PSet (XP
YP)                   'Plot point on screen
2320 Return
2400 Rem Test results
2410 If Abs(XNEW) + Abs(YNEW) > 1000000! Then T% = 2   'Unbounded
2430 GoSub 2900                'Calculate Lyapunov exponent
2460 If N >= NMAX Then T% = 2  'Strange attractor found
2470 If Abs(XNEW - X) + Abs(YNEW - Y) < .000001 Then T% = 2
2480 If N > 100 And L < .005 Then T% = 2   'Limit cycle
2490 DoEvents                  'Respond to user command
2510 X = XNEW                  'Update value of X
2520 Y = YNEW
2550 Return
2600 Rem Get coefficients
2650 O% = 2 + Int((OMAX% - 1) * Rnd)
2660 CODE$ = Chr$(59 + 4 * D% + O%)
2680 M% = 1: For I% = 1 To D%:  M% = M% * (O% + I%):  Next I%
2690      For I% = 1 To M%        'Construct CODE$
2700              GoSub 2800          'Shuffle random numbers
2710              CODE$ = CODE$ + Chr$(65 + Int(25 * RAN))
2720      Next I%
2730 For I% = 1 To M%          'Convert CODE$ to coefficient values
2740      A(I%) = (Asc(Mid$(CODE$
I% + 1
1)) - 77) / 10
2750 Next I%
2760 Return
2800 Rem Shuffle random numbers
2810 If V(0) = 0 Then For J% = 0 To 99: V(J%) = Rnd:  Next J%
2820 J% = Int(100 * RAN)
2830 RAN = V(J%)
2840 V(J%) = Rnd
2850 Return
2900 Rem Calculate Lyapunov exponent
2910 XSAVE = XNEW: YSAVE = YNEW:  X = XE:  Y = YE:  N = N - 1
2930 GoSub 1700                'Reiterate equations
2940 DLX = XNEW - XSAVE: DLY = YNEW - YSAVE
2960 DL2 = DLX * DLX + DLY * DLY
2970 If CSng(DL2) <= 0 Then GoTo 3070      'Don't divide by zero
2980      DF = 1000000000000# * DL2
2990      RS = 1 / Sqr(DF)
3000      XE = XSAVE + RS * (XNEW - XSAVE):  YE = YSAVE + RS * (YNEW - YSAVE)
3020      XNEW = XSAVE:  YNEW = YSAVE
3030      If DF > 0 Then LSUM = LSUM + Log(DF):  NL = NL + 1
3040      L = .721347 * LSUM / NL
3070 Return
3100 Rem Resize the screen
3110 If D% = 1 Then YMIN = XMIN: YMAX = XMAX
3120 If XMAX - XMIN < .000001 Then XMIN = XMIN - .0000005: XMAX = XMAX + .0000005
3130 If YMAX - YMIN < .000001 Then YMIN = YMIN - .0000005: YMAX = YMAX + .0000005
3160 MX = .1 * (XMAX - XMIN): MY = .1 * (YMAX - YMIN)
3170 XL = XMIN - MX: XH = XMAX + MX:  YL = YMIN - MY:  YH = YMAX + MY
3180 Scale (XL
YL)-(XH
YH): Cls
3460 Return
End Sub

QuickBASIC for Apple Macintosh Systems

If you want to run the programs on an Apple Macintosh the easiest way is to use the Macintosh version of QuickBASIC. Unfortunately this BASIC (at least in version 1.0) is not very compatible with any of the IBM BASICs and it lacks many important and useful commands although most of the missing features (such as color) are available through BASIC calls to the Macintosh Toolbox. An alternate though probably equally difficult approach is to convert the C source listing in Appendix D for use with one of the C compilers available for the Macintosh.

The QuickBASIC for Macintosh version of the programs typically executes more slowly than those compiled with the IBM version of QuickBASIC. For example the program used to produce the data in Table 2.2 finds about 106 attractors per hour when compiled with the Macintosh version of QuickBASIC and run on a 25 MHz Macintosh IIci with a floating-point coprocessor and only 14 per hour when using the QuickBASIC interpreter on the same computer.

To get you started the listing PROG06QB.MAC is a QuickBASIC for Macintosh version of PROG06. You can use it as a starting point for converting the other programs in this book for use on the Macintosh. You can use the Apple File Exchange utility to transfer PROG06QB.MAC on the accompanying disk to a Macintosh with a high density (1.4 MB) disk drive.

PROG06QB.MAC Macintosh QuickBASIC Version of PROG06

1000 REM TWO-D MAP SEARCH  Macintosh QuickBASIC Ver 1.0 (c) 1993 by J. C. Sprott

1010 DEFDBL A-Z                 'Use double precision
1020 DIM XS(499)
A(504)
V(99)
1040 PREV% = 5                  'Plot versus fifth previous iterate
1050 NMAX = 11000               'Maximum number of iterations
1060 OMAX% = 2                  'Maximum order of polynomial
1070 D% = 2                     'Dimension of system
1100 SND% = 0                   'Turn sound off
1160 RANDOMIZE TIMER            'Reseed random number generator
1190 GOSUB 1300                 'Initialize
1200 GOSUB 1500                 'Set parameters
1210 GOSUB 1700                 'Iterate equations
1220 GOSUB 2100                 'Display results
1230 GOSUB 2400                 'Test results
1240 ON T% GOTO 1190
1200
1210
1250 CLS
1260 END
1300 REM Initialize
1320 WINDOW 1
"Strange Attractors"
(0
36)-(SYSTEM(5)
SYSTEM(6))
1
1350 MENU 2
0
1
"Options": MENU 2
1
SND% + 1
"Sound"
1360 WW = WINDOW(2) / 2: WH = WINDOW(3) / 2: CLS
1370 BUTTON 1
1
"Searching..."
(WW - 45
WH - 10) - (WW + 45
WH + 10)
1420 RETURN
1500 REM Set parameters
1510 X = .05                    'Initial condition
1520 Y = .05
1550 XE = X + .000001: YE = Y
1560 GOSUB 2600                 'Get coefficients
1570 T% = 3
1580 P% = 0: LSUM = 0: N = 0: NL = 0
1590 XMIN = 1000000!: XMAX = -XMIN: YMIN = XMIN: YMAX = XMAX
1630 RETURN
1700 REM Iterate equations
1720 XNEW = A(1) + X * (A(2) + A(3) * X + A(4) * Y)
1730 XNEW = XNEW + Y * (A(5) + A(6) * Y)
1830 YNEW = A(7) + X * (A(8) + A(9) * X + A(10) * Y)
1930 YNEW = YNEW + Y * (A(11) + A(12) * Y)
2020 N = N + 1
2030 RETURN
2100 REM Display results
2110 IF N < 100 OR N > 1000 THEN GOTO 2200
2120    IF X < XMIN THEN XMIN = X
2130    IF X > XMAX THEN XMAX = X
2140    IF Y < YMIN THEN YMIN = Y
2150    IF Y > YMAX THEN YMAX = Y
2200 IF N = 1000 THEN GOSUB 3100        'Resize the screen
2210 XS(P%) = X
2220 P% = (P% + 1) MOD 500
2230 I% = (P% + 500 - PREV%) MOD 500
2240 IF D% = 1 THEN XP = XS(I%): YP = XNEW ELSE XP = X: YP = Y
2250 IF N < 1000 OR XP <= XL OR XP >= XH OR YP <= YL OR YP >= YH THEN GOTO 2320
2300    PSET (WW * (XP - XL) / (XH - XL)
WH * (YH - YP) / (YH - YL))
2310    IF SND% = 1 THEN GOSUB 3500     'Produce sound
2320 RETURN
2400 REM Test results
2410 IF ABS(XNEW) + ABS(YNEW) > 1000000! THEN T% = 2    'Unbounded
2430 GOSUB 2900                 'Calculate Lyapunov exponent
2460 IF N >= NMAX THEN T% = 2   'Strange attractor found
2470 IF ABS(XNEW - X) + ABS(YNEW - Y) < .000001 THEN T% = 2
2480 IF N > 100 AND L < .005 THEN T% = 2    'Limit cycle
2490 Q$ = INKEY$: IF LEN(Q$) THEN GOSUB 3600        'Respond to user command
2500 IF MENU(0) = 2 AND MENU(1) = 1 THEN Q$ = "S": GOSUB 3600
2510 X = XNEW                   'Update value of X
2520 Y = YNEW
2550 RETURN
2600 REM Get coefficients
2650 O% = 2 + INT((OMAX% - 1) * RND)
2660 CODE$ = CHR$(59 + 4 * D% + O%)
2680 M% = 1: FOR I% = 1 TO D%: M% = M% * (O% + I%): NEXT I%
2690    FOR I% = 1 TO M%        'Construct CODE$
2700        GOSUB 2800          'Shuffle random numbers
2710        CODE$ = CODE$ + CHR$(65 + INT(25 * RAN))
2720    NEXT I%
2730 FOR I% = 1 TO M%           'Convert CODE$ to coefficient values
2740    A(I%) = (ASC(MID$(CODE$
I% + 1
1)) - 77) / 10
2750 NEXT I%
2760 RETURN
2800 REM Shuffle random numbers
2810 IF V(0) = 0 THEN FOR J% = 0 TO 99: V(J%) = RND: NEXT J%
2820 J% = INT(100 * RAN)
2830 RAN = V(J%)
2840 V(J%) = RND
2850 RETURN
2900 REM Calculate Lyapunov exponent
2910 XSAVE = XNEW: YSAVE = YNEW: X = XE: Y = YE: N = N - 1
2930 GOSUB 1700                 'Reiterate equations
2940 DLX = XNEW - XSAVE: DLY = YNEW - YSAVE
2960 DL2 = DLX * DLX + DLY * DLY
2970 IF CSNG(DL2) <= 0 THEN GOTO 3070       'Don't divide by zero
2980    DF = 1000000000000# * DL2
2990    RS = 1 / SQR(DF)
3000    XE = XSAVE + RS * (XNEW - XSAVE): YE = YSAVE + RS * (YNEW - YSAVE)
3020    XNEW = XSAVE: YNEW = YSAVE
3030    IF DF > 0 THEN LSUM = LSUM + LOG(DF): NL = NL + 1
3040    L = .721347 * LSUM / NL
3070 RETURN
3100 REM Resize the screen
3110 IF D% = 1 THEN YMIN = XMIN: YMAX = XMAX
3120 IF XMAX - XMIN < .000001 THEN XMIN = XMIN - .0000005: XMAX = XMAX + .0000005
3130 IF YMAX - YMIN < .000001 THEN YMIN = YMIN - .0000005: YMAX = YMAX + .0000005
3160 MX = .1 * (XMAX - XMIN): MY = .1 * (YMAX - YMIN)
3170 XL = XMIN - MX: XH = XMAX + MX: YL = YMIN - MY: YH = YMAX + MY
3180 WW = WINDOW(2): WH = WINDOW(3): BUTTON CLOSE 0: CLS
3460 RETURN
3500 REM Produce sound
3510 FREQ% = 220 * 2 ^ (CINT(36 * (XNEW - XL) / (XH - XL)) / 12)
3520 DUR = 1
3540 SOUND FREQ%
DUR: IF PLAY(0) THEN PLAY "MF"
3550 RETURN
3600 REM Respond to user command
3610 T% = 0
3630 IF ASC(Q$) > 96 THEN Q$ = CHR$(ASC(Q$) - 32)
3770 IF Q$ = "S" THEN SND% = (SND% + 1) MOD 2: T% = 3: MENU 2
1
SND% + 1
"Sound"
3800 RETURN

APPENDIX D

C Program Listing

This appendix contains a translation of the BASIC program PROG28.BAS in Appendix B into the C language. The C language is preferred by many programmers because of its efficiency economy and portability. However the language is relatively sparse and relies on machine-specific run-time libraries for most input and output. Although there is a C standard (ANSI C) many necessary extensions are incorporated into various C compilers. These extensions also differ from one platform to another.

The listing here should compile and run without modification under Microsoft QuickC version 2.5 on the IBM PC-compatible platform. Some changes will be required to run under other versions of C. The disk included with this book contains the Microsoft QuickC source listing in a file PROG28QC.C and a version PROG28TC.CPP that should compile and run with Borland TurboC++ version 3.0. The programs will compile using the small memory model with either compiler. However you will probably find that the C versions run at about the same speed as the Microsoft QuickBASIC or VisualBASIC for MS-DOS version.

The C versions of the program are fairly literal translations of the BASIC version. All variables are global and retain the same names as in the BASIC version. The BASIC subroutines have been converted into C functions whose names correspond to the BASIC line numbers. No variables are passed to or from any of the functions. The level of indentation is minimal. The program assumes the computer has VGA color graphics. The listing PROG28QC.C should compile to a fully functional program except for the fact that QuickC lacks a sound function.

PROG28QC.C Microsoft QuickC Version of PROG28.BAS

/* STRANGE ATTRACTOR PROGRAM  QuickC Ver 2.0 (c) 1993 by J. C. Sprott */

#include <dos.h>
#include <stdio.h>
#include <graph.h>
#include <math.h>
int PREV
OMAX
D
ODE
SND
PJT
TRD
FTH
SAV
T
WID
QM
P
TWOD;
int M
I
I1
I2
O
I3
I4
I5
J
WH
FREQ
C4
NC
C
RD
CY
BK;
int COLR[16];
char CODE[515]
Q;
char FAV[9] = "XDATA.DAT";
double NMAX
EPS
TWOPI
SEG
NE
X
Y
Z
W
XE
YE
ZE
WE
LSUM
N
NL;
double N1
N2
XMIN
XMAX
YMIN
YMAX
ZMIN
ZMAX
WMIN
WMAX
XNEW
YNEW;
double ZNEW
WNEW
XP
YP
RAN
XSAVE
YSAVE
ZSAVE
WSAVE
DLX
DLY
DLZ;
double DLW
DL2
DF
RS
L
MX
MY
XL
XH
YL
YH
XA
YA
ZA
TT
PT
TIA;
double XZ
YZ
DUR
D2MAX
DX
DY
DZ
DW
D2
F
TH
PH
XRT
XLT
HSF
AL;
double SINAL
COSAL
DUM
SW
SH;
double XS[500]
YS[500]
ZS[500]
WS[500]
A[505]
V[100]
XY[5]
XN[5];
union REGS regs;
FILE *F1
*F2
*F3;
main()
{
PREV = 5;                   /* Plot versus fifth previous iterate */
NMAX = 11000;               /* Maximum number of iterations */
OMAX = 5;                   /* Maximum order of polynomial */
D = 2;                      /* Dimension of system */
EPS = .1;                   /* Step size for ODE */
ODE = 0;                    /* System is map */
SND = 0;                    /* Turn sound off */
PJT = 0;                    /* Projection is planar */
TRD = 1;                    /* Display third dimension as shadow */
FTH = 2;                    /* Display fourth dimension as colors */
SAV = 0;                    /* Don't save any data */
TWOPI = 6.28318530717959;   /* A useful constant (2 pi) */
srand(time());              /* Reseed random number generator */
fun4200();                  /* Display menu screen */
T = 1;
if (Q == 'X') T = 0;        /* Exit immediately on command */
while (T) {
switch (T) {
case 1: fun1300();  /* Initialize */
case 2: fun1500();  /* Set parameters */
case 3: fun1700();  /* Iterate equations */
case 4: fun2100();  /* Display results */
case 5: fun2400();  /* Test results */
}
}
_clearscreen(_GCLEARSCREEN);    /* Erase screen */
_setvideomode(_DEFAULTMODE);    /* and restore video mode */
}
fun1300() /* Initialize */
{
_setvideomode(_VRES16COLOR);    /* Assume VGA graphics */
WID = 80;                   /* Number of text columns */
_clearscreen(_GCLEARSCREEN);
_settextposition(13
WID / 2 - 6);
printf("Searching...");
fun5600();                  /* Set colors */
if (QM == 2) {
NE = 0;
fclose(F1);
F1 = fopen("SA.DIC"
"a");
fclose(F1);
F1 = fopen("SA.DIC"
"r");
}
}
fun1500() /* Set parameters */
{
X = .05;                    /* Initial condition */
Y = .05;
Z = .05;
W = .05;
XE = X + .000001;
YE = Y;
ZE = Z;
WE = W;
fun2600();                  /* Get coefficients */
T = 3;
P = 0;
LSUM = 0;
N = 0;
NL = 0;
N1 = 0;
N2 = 0;
XMIN = 1000000;
XMAX = -XMIN;
YMIN = XMIN;
YMAX = XMAX;
ZMIN = XMIN;
ZMAX = XMAX;
WMIN = XMIN;
WMAX = XMAX;
TWOD = _rotl(1
D);
}
fun1700() /* Iterate equations */
{
if (ODE > 1)
fun6200();              /* Special function */
else {
M = 1;
XY[1] = X;
XY[2] = Y;
XY[3] = Z;
XY[4] = W;
for (I = 1; I <= D; I++) {
XN[I] = A[M];
M = M + 1;
for (I1 = 1; I1 <= D; I1++) {
XN[I] = XN[I] + A[M] * XY[I1];
M = M + 1;
for (I2 = I1; I2 <= D; I2++) {
XN[I] = XN[I] + A[M] * XY[I1] * XY[I2];
M = M + 1;
for (I3 = I2; O > 2 && I3 <= D; I3++) {
XN[I] = XN[I] + A[M] * XY[I1] * XY[I2] * XY[I3];
M = M + 1;
for (I4 = I3; O > 3 && I4 <= D; I4++) {
XN[I] = XN[I] + A[M] * XY[I1] * XY[I2] * XY[I3] * XY[I4];
M = M + 1;
for (I5 = I4; O > 4 && I5 <= D; I5++) {
XN[I] = XN[I] + A[M] * XY[I1] * XY[I2] * XY[I3] * XY[I4] * XY[I5];
M = M + 1;
}}}}}
if (ODE == 1) XN[I] = XY[I] + EPS * XN[I];
}
XNEW = XN[1];
YNEW = XN[2];
ZNEW = XN[3];
WNEW = XN[4];
}
N = N + 1;
M = M - 1;
}
fun2100() /* Display results */
{
if (N >= 100 && N <= 1000) {
if (X < XMIN) XMIN = X;
if (X > XMAX) XMAX = X;
if (Y < YMIN) YMIN = Y;
if (Y > YMAX) YMAX = Y;
if (Z < ZMIN) ZMIN = Z;
if (Z > ZMAX) ZMAX = Z;
if (W < WMIN) WMIN = W;
if (W > WMAX) WMAX = W;
}
if ((int)N == 1000) fun3100();  /* Resize the screen */
XS[P] = X;
YS[P] = Y;
ZS[P] = Z;
WS[P] = W;
P = (P + 1) % 500;
I = (P + 500 - PREV) % 500;
if (D == 1) {
XP = XS[I];
YP = XNEW;
}
else {
XP = X;
YP = Y;
}
if (N >= 1000 && XP > XL && XP < XH && YP > YL && YP < YH) {
if (PJT == 1) fun4100();    /* Project onto a sphere */
if (PJT == 2) fun6700();    /* Project onto a horizontal cylinder */
if (PJT == 3) fun6800();    /* Project onto a vertical cylinder */
if (PJT == 4) fun6900();    /* Project onto a torus */
fun5000();                  /* Plot point on screen */
if (SND == 1) fun3500();    /* Produce sound */
}
}
fun2400() /* Test results */
{
if (fabs(XNEW) + fabs(YNEW) + fabs(ZNEW) + fabs(WNEW) > 1000000) T = 2;
if (QM != 2) {              /* Speed up evaluation mode */
fun2900();              /* Calculate Lyapunov exponent */
fun3900();              /* Calculate fractal dimension */
if (QM == 0) {          /* Skip tests unless in search mode */
if (N >= NMAX) {    /* Strange attractor found */
T = 2;
fun4900();      /* Save attractor to disk file SA.DIC */
}
if (fabs(XNEW - X) + fabs(YNEW - Y) + fabs(ZNEW - Z) + fabs(WNEW - W) < .000001) T = 2;
if (N > 100 && L < .005) T = 2; /* Limit cycle */
}
}
if (kbhit()) Q = getch(); else Q = 0;
if (Q) fun3600();           /* Respond to user command */
if (SAV > 0) if (N > 1000 && N < 17001) fun7000();  /* Save data */
X = XNEW;                   /* Update value of X */
Y = YNEW;
Z = ZNEW;
W = WNEW;
}
fun2600() /* Get coefficients */
{
if (QM == 2) {              /* In evaluate mode */
fgets(CODE
515
F1);
if (feof(F1)) {
QM = 0;
fun6000();          /* Update SA.DIC file */
}
else {
fun4700();          /* Get dimension and order */
fun5600();          /* Set colors */
}
}
if (QM == 0) {              /* In search mode */
O = 2 + (int)floor((OMAX - 1) * (float)rand() / 32768.0);
CODE[0] = 59 + 4 * D + O + 8 * ODE;
if (ODE > 1) CODE[0] = 87 + ODE;
fun4700();              /* Get value of M */
for (I = 1; I <= M; I++) {  /* Construct CODE */
fun2800();           /* Shuffle random numbers */
CODE[I] = 65 + (int)floor(25 * RAN);
}
CODE[M + 1] = 0;
}
for (I = 1; I <= M; I++) {  /* Convert CODE to coefficient values */
A[I] = (CODE[I] - 77) / 10.0;
}      
}
fun2800() /* Shuffle random numbers */
{
if (V[0] == 0) for (J = 0; J <= 99; J++) {V[J] = (float)rand() / 32768.0;}
J = (int)floor(100 * RAN);
RAN = V[J];
V[J] = (float)rand() / 32768.0;
}
fun2900() /* Calculate Lyapunov exponent */
{
XSAVE = XNEW;
YSAVE = YNEW;
ZSAVE = ZNEW;
WSAVE = WNEW;
X = XE;
Y = YE;
Z = ZE;
W = WE;
N = N - 1;
fun1700();                  /* Reiterate equations */
DLX = XNEW - XSAVE;
DLY = YNEW - YSAVE;
DLZ = ZNEW - ZSAVE;
DLW = WNEW - WSAVE;
DL2 = DLX * DLX + DLY * DLY + DLZ * DLZ + DLW * DLW;
if (DL2 > 0) {              /* Check for division by zero */
DF = 1E12 * DL2;
RS = 1 / sqrt(DF);
XE = XSAVE + RS * (XNEW - XSAVE);
YE = YSAVE + RS * (YNEW - YSAVE);
ZE = ZSAVE + RS * (ZNEW - ZSAVE);
WE = WSAVE + RS * (WNEW - WSAVE);
XNEW = XSAVE;
YNEW = YSAVE;
ZNEW = ZSAVE;
WNEW = WSAVE;
LSUM = LSUM + log(DF);
NL = NL + 1;
L = .721347 * LSUM / NL;
if (ODE == 1 || ODE == 7) L = L / EPS;
if (N > 1000 && (int)N % 10 == 0) {
_settextposition(1
WID - 4);
printf("%5.2f"
L);
}
}
}
fun3100() /* Resize the screen */
{
_setcolor(WH);
if (D == 1) {
YMIN = XMIN;
YMAX = XMAX;
}
if (XMAX - XMIN < .000001) {
XMIN = XMIN - .0000005;
XMAX = XMAX + .0000005;
}
if (YMAX - YMIN < .000001) {
YMIN = YMIN - .0000005;
YMAX = YMAX + .0000005;
}
if (ZMAX - ZMIN < .000001) {
ZMIN = ZMIN - .0000005;
ZMAX = ZMAX + .0000005;
}
if (WMAX - WMIN < .000001) {
WMIN = WMIN - .0000005;
WMAX = WMAX + .0000005;
}
MX = .1 * (XMAX - XMIN);
MY = .1 * (YMAX - YMIN);
XL = XMIN - MX;
XH = XMAX + MX;
YL = YMIN - MY;
YH = YMAX + 1.5 * MY;
SW = 640 / (XH - XL);
SH = 480 / (YL - YH);
_setvieworg((short)(-SW * XL)
(short)(-SH * YH));
_clearscreen(_GCLEARSCREEN);
YH = YH - .5 * MY;
XA = (XL + XH) / 2;
YA = (YL + YH) / 2;
if (D > 2) {
ZA = (ZMAX + ZMIN) / 2;
if (TRD == 1) {
_setcolor(COLR[1]);
_rectangle_w(_GFILLINTERIOR
SW * XL
SH * YL
SW * XH
SH * YH);
fun5400();          /* Plot background grid */
}
if (TRD == 4) {
_setcolor(WH);
_rectangle_w(_GFILLINTERIOR
SW * XL
SH * YL
SW * XH
SH * YH);
}
if (TRD == 5) {
_moveto_w(SW * XA
SH * YL);
_lineto_w(SW * XA
SH * YH);
}
if (TRD == 6) {
for (I = 1; I <= 3; I++) {
XP = XL + I * (XH - XL) / 4;
_moveto_w(SW * XP
SH * YL);
_lineto_w(SW * XP
SH * YH);
YP = YL + I * (YH - YL) / 4;
_moveto_w(SW * XL
SH * YP);
_lineto_w(SW * XH
SH * YP);
}
}
}
if (PJT != 1) _rectangle_w(_GBORDER
SW * XL + 1
SH * YL - 1
SW * XH - 1
SH * YH + 1);
if (PJT == 1 && TRD < 5) _ellipse_w(_GBORDER
SW * XL - SH * (YH - YL) / 6
SH * YH
SW * XH + SH * (YH - YL) / 6
SH * YL);
TT = 3.1416 / (XMAX - XMIN);
PT = 3.1416 / (YMAX - YMIN);
if (QM == 2) {              /* In evaluate mode */
_settextposition(1
1);
printf("<Space Bar>: Discard       <Enter>: Save");
if (WID >= 80) {
_settextposition(1
49);
printf("<Esc>: Exit");
_settextposition(1
69);
printf("%d K left"
(int)((filelength(fileno(F1)) - ftell(F1)) / 1024.0));
}}
else {
_settextposition(1
1);
if (strlen(CODE) < WID - 18)
_outtext(CODE);
else {
printf("%*.*s..."
WID - 23
WID - 23
CODE);
}
_settextposition(1
WID - 17);
printf("F =");
_settextposition(1
WID - 7);
printf("L = ");
}
TIA = .05;                  /* Tangent of illumination angle */
XZ = -TIA * (XMAX - XMIN) / (ZMAX - ZMIN);
YZ = TIA * (YMAX - YMIN) / (ZMAX - ZMIN);
}
fun3500() /* Produce sound */
{
FREQ = 220 * pow(2
(int)(36 * (XNEW - XL) / (XH - XL)) / 12.0);
DUR = 1;
if (D > 1) DUR = pow(2
(int)floor(.5 * (YH - YL) / (YNEW - 9 * YL / 8 + YH / 8)));
/* A sound statement should be placed here */
}
fun3600() /* Respond to user command */
{
if (Q > 96) Q = Q - 32;     /* Convert to upper case */
if (QM == 2) fun5800();     /* Process evaluation command */
if (strchr("ACDEHINPRSVX"
Q) == 0) fun4200();  /* Display menu screen */
if (Q == 'A') {
T = 1;
QM = 0;
}
if (ODE > 1) D = ODE + 5;
if (ODE == 1) D = D + 2;
if (Q == 'C') if (N > 999) N = 999;
if (Q == 'D') {
D = 1 + D % 12;
T = 1;
}
if (D > 6) {
ODE = D - 5;
D = 4;
}
else {
if (D > 4) {
ODE = 1;
D = D - 2;
}
else ODE = 0;
}
if (Q == 'E') {
T = 1;
QM = 2;
}
if (Q == 'H') {
FTH = (FTH + 1) % 3;
T = 3;
if (N > 999) {
N = 999;
fun5600();          /* Set colors */
}
}
if (Q == 'I') {
if (T != 1) {
_setvideomode(_TEXTC80);
_settextcolor(15);
_setbkcolor(1L);
_clearscreen(_GCLEARSCREEN);
printf("Code? ");
I = 0;
CODE[0] = 0;
do {
CODE[I] = getche();
if (CODE[I] == 8 && I >= 0) I = I - 2;
if (CODE[I] == 27) {
I = 0;
CODE[I] = 13;
}
}
while (CODE[I++] != 13 && I < 506);
CODE[I - 1] = 0;
if (CODE[0] == 0) {
Q = ' ';
_clearscreen(_GCLEARSCREEN);}
else {
T = 1;
QM = 1;
fun4700();
}
}
}
if (Q == 'N') {
NMAX = 10 * (NMAX - 1000) + 1000;
if (NMAX > 1E10) NMAX = 2000;
}
if (Q == 'P') {
PJT = (PJT + 1) % 5;
T = 3;
if (N > 999) N = 999;
}
if (Q == 'R') {
TRD = (TRD + 1) % 7;
T = 3;
if (N > 999) {
N = 999;
fun5600();          /* Get dimension and order */
}
}
if (Q == 'S') {
SND = (SND + 1) % 2;
T = 3;
}
if (Q == 'V') {
SAV = (SAV + 1) % 5;
FAV[0] = 87 + SAV % 4;
T = 3;
if (N > 999) N = 999;
}
if (Q == 'X') T = 0;
}
fun3900() /* Calculate fractal dimension */
{
if (N >= 1000) {            /* Wait for transient to settle */
if ((int)N == 1000) {
D2MAX = pow(XMAX - XMIN
2);
D2MAX = D2MAX + pow(YMAX - YMIN
2);
D2MAX = D2MAX + pow(ZMAX - ZMIN
2);
D2MAX = D2MAX + pow(WMAX - WMIN
2);
}
J = (P + 1 + (int)floor(480 * (float)rand() / 32768.0)) % 500;
DX = XNEW - XS[J];
DY = YNEW - YS[J];
DZ = ZNEW - ZS[J];
DW = WNEW - WS[J];
D2 = DX * DX + DY * DY + DZ * DZ + DW * DW;
if (D2 < .001 * TWOD * D2MAX) N2 = N2 + 1;
if (D2 <= .00001 * TWOD * D2MAX) {
N1 = N1 + 1;
F = .434294 * log(N2 / (N1 - .5));
_settextposition(1
WID - 14);
printf("%5.2f"
F);
}
}
}
fun4100() /* Project onto a sphere */
{
TH = TT * (XMAX - XP);
PH = PT * (YMAX - YP);
XP = XA + .36 * (XH - XL) * cos(TH) * sin(PH);
YP = YA + .5 * (YH - YL) * cos(PH);
}
fun4200() /* Display menu screen */
{
_setvideomode(_TEXTC80);
_settextcolor(15);
_setbkcolor(1L);
regs.h.ah = 1;
regs.h.ch = 1;
regs.h.cl = 0;
int86(16
&regs
&regs);    /* Turn cursor off */
_clearscreen(_GCLEARSCREEN);
while (Q == 0 || strchr("AEIX"
Q) == 0) {
_settextposition(1
27);
printf("STRANGE ATTRACTOR PROGRAM\n");
printf("%26cIBM PC QuickC Version 2.0\n"
' ');
printf("%26c(c) 1993 by J. C. Sprott\n"
' ');
printf("\n");
printf("\n");
printf("%26cA: Search for attractors\n"
' ');
printf("%26cC: Clear screen and restart\n"
' ');
if (ODE > 1) {
printf("%26cD: System is 4-D special map %c \n"
' '
87 + ODE);}
else {
printf("%26cD: System is %d-D polynomial "
' '
D);
if (ODE == 1) printf("ODE\n"); else printf("map\n");
}
printf("%26cE: Evaluate attractors\n"
' ');
printf("%26cH: Fourth dimension is "
' ');
if (FTH == 0) printf("projection\n");
if (FTH == 1) printf("bands     \n");
if (FTH == 2) printf("colors    \n");
printf("%26cI: Input code from keyboard\n"
' ');
printf("%26cN: Number of iterations is 10^%1.0f\n"
' '
log10(NMAX - 1000));
printf("%26cP: Projection is "
' ');
if (PJT == 0) printf("planar   \n");
if (PJT == 1) printf("spherical\n");
if (PJT == 2) printf("horiz cyl\n");
if (PJT == 3) printf("vert cyl \n");
if (PJT == 4) printf("toroidal \n");
printf("%26cR: Third dimension is "
' ');
if (TRD == 0) printf("projection\n");
if (TRD == 1) printf("shadow    \n");
if (TRD == 2) printf("bands     \n");
if (TRD == 3) printf("colors    \n");
if (TRD == 4) printf("anaglyph  \n");
if (TRD == 5) printf("stereogram\n");
if (TRD == 6) printf("slices    \n");
printf("%26cS: Sound is "
' ');
if (SND == 0) printf("off\n");
if (SND == 1) printf("on \n");
printf("%26cV: "
' ');
if (SAV == 0) printf("No data will be saved       \n");
if (SAV > 0) printf("%c will be saved in %cDATA.DAT\n"
FAV[0]
FAV[0]);
printf("%26cX: Exit program"
' ');
if (kbhit()) Q = getch(); else Q = 0;
if (Q) fun3600();       /* Respond to user command */
}
}
fun4700() /* Get dimension and order */
{
D = 1 + (int)floor((CODE[0] - 65) / 4);
if (D > 6) {
ODE = CODE[0] - 87;
D = 4;
fun6200();              /* Special function */
}
else {
if (D > 4) {
D = D - 2;
ODE = 1;
}
else ODE = 0;
O = 2 + (CODE[0] - 65) % 4;
M = 1;
for (I = 1; I <= D; I++) {M = M * (O + I);}
if (D > 2) for (I = 3; I <= D; I++) {M = M / (I - 1);}
}
if (strlen(CODE) != M + 1 && QM == 1) {
printf("\a");            /* Illegal code warning */
while (strlen(CODE) < M + 1) strcat(CODE
"M");
if (strlen(CODE) > M + 1) CODE[M + 1] = 0;
}
}
fun4900() /* Save attractor to disk file SA.DIC */
{
F1 = fopen("SA.DIC"
"a");
fprintf(F1
"%s%5.2f%5.2f\n"
CODE
F
L);
fclose(F1);
}
fun5000() /* Plot point on screen */
{
C4 = WH;
if (D > 3) {
if (FTH == 1) if ((int)floor(30 * (W - WMIN) / (WMAX - WMIN)) % 2) return(0);
if (FTH == 2) C4 = 1 + (int)floor(NC * (W - WMIN) / (WMAX - WMIN) + NC) % NC;
}
if (D < 3) {                /* Skip 3-D stuff */
_setpixel_w(SW * XP
SH * YP);
return(0);
}
if (TRD == 0) {
_setcolor(C4);
_setpixel_w(SW * XP
SH * YP);
}
if (TRD == 1) {
if (D > 3 && FTH == 2) {
_setcolor(C4);
_setpixel_w(SW * XP
SH * YP);
}
else {
C = _getpixel_w(SW * XP
SH * YP);
if (C == COLR[2]) {
_setcolor(COLR[3]);
_setpixel_w(SW * XP
SH * YP);}
else {
if (C != COLR[3]) {
_setcolor(COLR[2]);
_setpixel_w(SW * XP
SH * YP);
}
}
}
XP = XP - XZ * (Z - ZMIN);
YP = YP - YZ * (Z - ZMIN);
if (_getpixel_w(SW * XP
SH * YP) == COLR[1]) {
_setcolor(0);
_setpixel_w(SW * XP
SH * YP);
}
}
if (TRD == 2) {
if (D > 3 && FTH == 2 && ((int)floor(15 * (Z - ZMIN) / (ZMAX - ZMIN) + 2) % 2) == 1) {
_setcolor(C4);}
else {
C = COLR[(int)floor(60 * (Z - ZMIN) / (ZMAX - ZMIN) + 4) % 4];
_setcolor(C);
}
_setpixel_w(SW * XP
SH * YP);
}
if (TRD == 3) {
_setcolor(COLR[(int)floor(NC * (Z - ZMIN) / (ZMAX - ZMIN) + NC) % NC]);
_setpixel_w(SW * XP
SH * YP);
}
if (TRD == 4) {
XRT = XP + XZ * (Z - ZA);
C = _getpixel_w(SW * XRT
SH * YP);
if (C == WH) {
_setcolor(RD);
_setpixel_w(SW * XRT
SH * YP);
}
if (C == CY) {
_setcolor(BK);
_setpixel_w(SW * XRT
SH * YP);
}
XLT = XP - XZ * (Z - ZA);
C = _getpixel_w(SW * XLT
SH * YP);
if (C == WH) {
_setcolor(CY);
_setpixel_w(SW * XLT
SH * YP);
}
if (C == RD) {
_setcolor(BK);
_setpixel_w(SW * XLT
SH * YP);
}
}
if (TRD == 5) {
HSF = 2;                /* Horizontal scale factor */
XRT = XA + (XP + XZ * (Z - ZA) - XL) / HSF;
_setcolor(C4);
_setpixel_w(SW * XRT
SH * YP);
XLT = XA + (XP - XZ * (Z - ZA) - XH) / HSF;
_setcolor(C4);
_setpixel_w(SW * XLT
SH * YP);
}
if (TRD == 6) {
DZ = (15 * (Z - ZMIN) / (ZMAX - ZMIN) + .5) / 16;
XP = (XP - XL + ((int)floor(16 * DZ) % 4) * (XH - XL)) / 4 + XL;
YP = (YP - YL + (3 - (int)floor(4 * DZ) % 4) * (YH - YL)) / 4 + YL;
_setcolor(C4);
_setpixel_w(SW * XP
SH * YP);
}
}
fun5400() /* Plot background grid */
{
_setcolor(0);
for (I = 0; I <= 15; I++) { /* Draw 15 vertical grid lines */
XP = XMIN + I * (XMAX - XMIN) / 15;
_moveto_w(SW * XP
SH * YMIN);
_lineto_w(SW * XP
SH * YMAX);
}      
for (I = 0; I <= 10; I++) { /* Draw 10 horizontal grid lines */
YP = YMIN + I * (YMAX - YMIN) / 10;
_moveto_w(SW * XMIN
SH * YP);
_lineto_w(SW * XMAX
SH * YP);
}
}
fun5600() /* Set colors */
{
NC = 15;                    /* Number of colors */
COLR[0] = 0;
COLR[1] = 8;
COLR[2] = 7;
COLR[3] = 15;
if (TRD == 3 || (D > 3 && FTH == 2 && TRD != 1)) {
for (I = 0; I <= NC; I++) COLR[I] = I + 1;
}
WH = 15;
BK = 8;
RD = 12;
CY = 11;
}
fun5800() /* Process evaluation command */
{
if (Q == ' ') {
T = 2;
NE = NE + 1;
_clearscreen(_GCLEARSCREEN);
}
if (Q == 13) {
T = 2;
NE = NE + 1;
_clearscreen(_GCLEARSCREEN);
fun5900();              /* Save favorite attractors to disk */
}
if (Q == 27) {
_clearscreen(_GCLEARSCREEN);
fun6000();              /* Update SA.DIC file */
Q = ' ';
QM = 0;
}
else {
if (strchr("CHNPRVS"
Q) == 0) Q = 0;
}
}
fun5900() /* Save favorite attractors to disk file FAVORITE.DIC */
{
F2 = fopen("FAVORITE.DIC"
"a");
fprintf(F2
CODE);
fclose(F2);
}
fun6000() /* Update SA.DIC file */
{
_settextposition(11
9);
printf("Evaluation complete\n");
_settextposition(12
8);
printf(" %d cases evaluated"
(int)NE);
F2 = fopen("SATEMP.DIC"
"w");
if (QM == 2) fprintf(F2
CODE);
while (feof(F1) == 0) {
fgets(CODE
515
F1);
if (feof(F1) == 0) fprintf(F2
CODE);
}
fcloseall();
remove("SA.DIC");
rename("SATEMP.DIC"
"SA.DIC");
}
fun6200() /* Special function definitions */
{
ZNEW = X * X + Y * Y;       /* Default 3rd and 4th dimension */
WNEW = (N - 100) / 900;
if (N > 1000) WNEW = (N - 1000) / (NMAX - 1000);
if (ODE == 2) {
M = 10;
XNEW = A[1] + A[2] * X + A[3] * Y + A[4] * fabs(X) + A[5] * fabs(Y);
YNEW = A[6] + A[7] * X + A[8] * Y + A[9] * fabs(X) + A[10] * fabs(Y);
}
if (ODE == 3) {
M = 14;
XNEW = A[1] + A[2] * X + A[3] * Y + ((int)(A[4] * X + .5) & (int)(A[5] * Y + .5)) + ((int)(A[6] * X + .5) | (int)(A[7] * Y + .5));
YNEW = A[8] + A[9] * X + A[10] * Y + ((int)(A[11] * X + .5) & (int)(A[12] * Y + .5)) + ((int)(A[13] * X + .5) | (int)(A[14] * Y + .5));
}
if (ODE == 4) {
M = 14;
XNEW = A[1] + A[2] * X + A[3] * Y + A[4] * pow(fabs(X)
A[5]) + A[6] * pow(fabs(Y)
A[7]);
YNEW = A[8] + A[9] * X + A[10] * Y + A[11] * pow(fabs(X)
A[12]) + A[13] * pow(fabs(Y)
A[14]);
}
if (ODE == 5) {
M = 18;
XNEW = A[1] + A[2] * X + A[3] * Y + A[4] * sin(A[5] * X + A[6]) + A[7] * sin(A[8] * Y + A[9]);
YNEW = A[10] + A[11] * X + A[12] * Y + A[13] * sin(A[14] * X + A[15]) + A[16] * sin(A[17] * Y + A[18]);
}
if (ODE == 6) {
M = 6;
if (N < 2) {
AL = TWOPI / (13 + 10 * A[6]);
SINAL = sin(AL);
COSAL = cos(AL);
}
DUM = X + A[2] * sin(A[3] * Y + A[4]);
XNEW = 10 * A[1] + DUM * COSAL + Y * SINAL;
YNEW = 10 * A[5] - DUM * SINAL + Y * COSAL;
}
if (ODE == 7) {
M = 9;
XNEW = X + EPS * A[1] * Y;
YNEW = Y + EPS * (A[2] * X + A[3] * X * X * X + A[4] * X * X * Y + A[5] * X * Y * Y + A[6] * Y + A[7] * Y * Y * Y + A[8] * sin(Z));
ZNEW = Z + EPS * (A[9] + 1.3);
if (ZNEW > TWOPI) ZNEW = ZNEW - TWOPI;
}
}
fun6700() /* Project onto a horizontal cylinder */
{
PH = PT * (YMAX - YP);
YP = YA + .5 * (YH - YL) * cos(PH);
}
fun6800() /* Project onto a vertical cylinder */
{
TH = TT * (XMAX - XP);
XP = XA + .5 * (XH - XL) * cos(TH);
}
fun6900() /* Project onto a torus (unity aspect ratio) */
{
TH = TT * (XMAX - XP);
PH = 2 * PT * (YMAX - YP);
XP = XA + .18 * (XH - XL) * (1 + cos(TH)) * sin(PH);
YP = YA + .25 * (YH - YL) * (1 + cos(TH)) * cos(PH);
}
fun7000() /* Save data */
{
if ((int)N == 1001) {
fclose(F3);
F3 = fopen(FAV
"w");
}
if (SAV == 1) DUM = XNEW;
if (SAV == 2) DUM = YNEW;
if (SAV == 3) DUM = ZNEW;
if (SAV == 4) DUM = WNEW;
fprintf(F3
"%f\n"
DUM);
}

APPENDIX E

Summary of Equations

This appendix contains a complete list in all its gory detail of the equations solved by the program to produce the attractors in this book. For simplicity the subscripts n+1 and n have been omitted on the variables X Y Z and W. If it serves no other purpose this appendix vividly illustrates the power of programming languages in expressing and evaluating lengthy formulas! It is worth emphasizing that the attractors that come from simple equations are every bit as interesting and beautiful as those that come from complicated equations.

Case A: D = 1 O = 2 M = 3

X = a1 + a2X + a3X2

Case B: D = 1 O = 3 M = 4

X = a1 + a2X + a3X2 + a4X3

Case C: D = 1 O = 4 M = 5

X = a1 + a2X + a3X2 + a4X3 + a5X4

Case D: D = 1 O = 5 M = 6

X = a1 + a2X + a3X2 + a4X3 + a5X4 + a6X5

Case E: D = 2 O = 2 M = 12

X = a1 + a2X + a3X2 + a4XY + a5Y + a6Y2

Y = a7 + a8X + a9X2 + a10XY + a11Y + a12Y2

Case F: D = 2 O = 3 M = 20

X = a1 + a2X + a3X2 + a4X3 + a5X2Y + a6XY + a7XY2 + a8Y + a9Y2 + a10Y3

Y = a11 + a12X + a13X2 + a14X3 + a15X2Y + a16XY + a17XY2 + a18Y + a19Y2 + a20Y3

Case G: D = 2 O = 4 M = 30

X = a1 + a2X + a3X2 + a4X3 + a5X4 + a6X3Y + a7X2Y + a8X2Y2 + a9XY + a10XY2 + a11XY3 + a12Y + a13Y2 + a14Y3 + a15Y4

Y = a16 + a17X + a18X2 + a19X3 + a20X4 + a21X3Y + a22X2Y + a23X2Y2 + a24XY + a25XY2 + a26XY3 + a27Y + a28Y2 + a29Y3 + a30Y4

Case H: D = 2 O = 5 M = 42

X = a1 + a2X + a3X2 + a4X3 + a5X4 + a6X5 + a7X4Y + a8X3Y + a9X3Y2 + a10X2Y + a11X2Y2 + a12X2Y3 + a13XY + a14XY2 + a15XY3 + a16XY4 + a17Y + a18Y2 + a19Y3 + a20Y4 + a21Y5

Y = a22 + a23X + a24X2 + a25X3 + a26X4 + a27X5 + a28X4Y + a29X3Y + a30X3Y2 + a31X2Y + a32X2Y2 + a33X2Y3 + a34XY + a35XY2 + a36XY3 + a37XY4 + a38Y + a39Y2 + a40Y3 + a41Y4 + a42Y5

Case I: D = 3 O = 2 M = 30

X = a1 + a2X + a3X2 + a4XY + a5XZ + a6Y + a7Y2 + a8YZ + a9Z + a10Z2

Y = a11 + a12X + a13X2 + a14XY + a15XZ + a16Y + a17Y2 + a18YZ + a19Z + a20Z2
Z = a21 + a22X + a23X2 + a24XY + a25XZ + a26Y + a27Y2 + a28YZ + a29Z + a30Z2

Case J: D = 3 O = 3 M = 60

X = a1 + a2X + a3X2 + a4X3 + a5X2Y + a6X2Z + a7XY + a8XY2 + a9XYZ + a10XZ + a11XZ2 + a12Y + a13Y2 + a14Y3 + a15Y2Z + a16YZ + a17YZ2 + a18Z + a19Z2 + a20Z3

Y = a21 + a22X + a23X2 + a24X3 + a25X2Y + a26X2Z + a27XY + a28XY2 + a29XYZ + a30XZ + a31XZ2 + a32Y + a33Y2 + a34Y3 + a35Y2Z + a36YZ + a37YZ2 + a38Z + a39Z2 + a40Z3
Z = a41 + a42X + a43X2 + a44X3 + a45X2Y + a46X2Z + a47XY + a48XY2 + a49XYZ + a50XZ + a51XZ2 + a52Y + a53Y2 + a54Y3 + a55Y2Z + a56YZ + a57YZ2 + a58Z + a59Z2 + a60Z3

Case K: D = 3 O = 4 M = 105

X = a1 + a2X + a3X2 + a4X3 + a5X4 + a6X3Y + a7X3Z + a8X2Y + a9X2Y2 + a10X2YZ + a11X2Z + a12X2Z2 + a13XY + a14XY2 + a15XY3 + a16XY2Z + a17XYZ + a18XYZ2 + a19XZ + a20XZ2 + a21XZ3 + a22Y + a23Y2 + a24Y3 + a25Y4 + a26Y3Z + a27Y2Z + a28Y2Z2 + a29YZ + a30YZ2 + a31YZ3 + a32Z + a33Z2 + a34Z3 + a35Z4

Y = a36 + a37X + a38X2 + a39X3 + a40X4 + a41X3Y + a42X3Z + a43X2Y + a44X2Y2 + a45X2YZ + a46X2Z + a47X2Z2 + a48XY + a49XY2 + a50XY3 + a51XY2Z + a52XYZ + a53XYZ2 + a54XZ + a55XZ2 + a56XZ3 + a57Y + a58Y2 + a59Y3 + a60Y4 + a61Y3Z + a62Y2Z + a63Y2Z2 + a64YZ + a65YZ2 + a66YZ3 + a67Z + a68Z2 + a69Z3 + a70Z4
Z = a71 + a72X + a73X2 + a74X3 + a75X4 + a76X3Y + a77X3Z + a78X2Y + a79X2Y2 + a80X2YZ + a81X2Z + a82X2Z2 + a83XY + a84XY2 + a85XY3 + a86XY2Z + a87XYZ + a88XYZ2 + a89XZ + a90XZ2 + a91XZ3 + a92Y + a93Y2 + a94Y3 + a95Y4 + a96Y3Z + a97Y2Z + a98Y2Z2 + a99YZ + a100YZ2 + a101YZ3 + a102Z + a103Z2 + a104Z3 + a105Z4

Case L: D = 3 O = 5 M = 168

X = a1 + a2X + a3X2 + a4X3 + a5X4 + a6X5 + a7X4Y + a8X4Z + a9X3Y + a10X3Y2 + a11X3YZ + a12X3Z + a13X3Z2 + a14X2Y + a15X2Y2 + a16X2Y3 + a17X2Y2Z + a18X2YZ + a19X2YZ2 + a20X2Z + a21X2Z2 + a22X2Z3 + a23XY + a24XY2 + a25XY3 + a26XY4 + a27XY3Z + a28XY2Z + a29XY2Z2 + a30XYZ + a31XYZ2 + a32XYZ3 + a33XZ + a34XZ2 + a35XZ3 + a36XZ4 + a37Y + a38Y2 + a39Y3 + a40Y4 + a41Y5 + a42Y4Z + a43Y3Z + a44Y3Z2 + a45Y2Z + a46Y2Z2 + a47Y2Z3 + a48YZ + a49YZ2 + a50YZ3 + a51YZ4 + a52Z + a53Z2 + a54Z3 + a55Z4 + a56Z5

Y = a57 + a58X + a59X2 + a60X3 + a61X4 + a62X5 + a63X4Y + a64X4Z + a65X3Y + a66X3Y2 + a67X3YZ + a68X3Z + a69X3Z2 + a70X2Y + a71X2Y2 + a72X2Y3 + a73X2Y2Z + a74X2YZ + a75X2YZ2 + a76X2Z + a77X2Z2 + a78X2Z3 + a79XY + a80XY2 + a81XY3 + a82XY4 + a83XY3Z + a84XY2Z + a85XY2Z2 + a86XYZ + a87XYZ2 + a88XYZ3 + a89XZ + a90XZ2 + a91XZ3 + a92XZ4 + a93Y + a94Y2 + a95Y3 + a96Y4 + a97Y5 + a98Y4Z + a99Y3Z + a100Y3Z2 + a101Y2Z + a102Y2Z2 + a103Y2Z3 + a104YZ + a105YZ2 + a106YZ3 + a107YZ4 + a108Z + a109Z2 + a110Z3 + a111Z4 + a112Z5
Z = a113 + a114X + a115X2 + a116X3 + a117X4 + a118X5 + a119X4Y + a120X4Z + a121X3Y + a122X3Y2 + a123X3YZ + a124X3Z + a125X3Z2 + a126X2Y + a127X2Y2 + a128X2Y3 + a129X2Y2Z + a130X2YZ + a131X2YZ2 + a132X2Z + a133X2Z2 + a134X2Z3 + a135XY + a136XY2 + a137XY3 + a138XY4 + a139XY3Z + a140XY2Z + a141XY2Z2 + a142XYZ + a143XYZ2 + a144XYZ3 + a145XZ + a146XZ2 + a147XZ3 + a148XZ4 + a149Y + a150Y2 + a151Y3 + a152Y4 + a153Y5 + a154Y4Z + a155Y3Z + a156Y3Z2 + a157Y2Z + a158Y2Z2 + a159Y2Z3 + a160YZ + a161YZ2 + a162YZ3 + a163YZ4 + a164Z + a165Z2 + a166Z3 + a167Z4 + a168Z5

Case M: D = 4 O = 2 M = 60

X = a1 + a2X + a3X2 + a4XY + a5XZ + a6XW + a7Y + a8Y2 + a9YZ + a10YW + a11Z + a12Z2 + a13ZW + a14W + a15W2

Y = a16 + a17X + a18X2 + a19XY + a20XZ + a21XW + a22Y + a23Y2 + a24YZ + a25YW + a26Z + a27Z2 + a28ZW + a29W + a30W2
Z = a31 + a32X + a33X2 + a34XY + a35XZ + a36XW + a37Y + a38Y2 + a39YZ + a40YW + a41Z + a42Z2 + a43ZW + a44W + a45W2
W = a46 + a47X + a48X2 + a49XY + a50XZ + a51XW + a52Y + a53Y2 + a54YZ + a55YW + a56Z + a57Z2 + a58ZW + a59W + a60W2

Case N: D = 4 O = 3 M = 140

X = a1 + a2X + a3X2 + a4X3 + a5X2Y + a6X2Z + a7X2W + a8XY + a9XY2 + a10XYZ + a11XYW + a12XZ + a13XZ2 + a14XZW + a15XW + a16XW2 + a17Y + a18Y2 + a19Y3 + a20Y2Z + a21Y2W + a22YZ + a23YZ2 + a24YZW + a25YW + a26YW2 + a27Z + a28Z2 + a29Z3 + a30Z2W + a31ZW + a32ZW2 + a33W + a34W2 + a35W3

Y = a36 + a37X + a38X2 + a39X3 + a40X2Y + a41X2Z + a42X2W + a43XY + a44XY2 + a45XYZ + a46XYW + a47XZ + a48XZ2 + a49XZW + a50XW + a51XW2 + a52Y + a53Y2 + a54Y3 + a55Y2Z + a56Y2W + a57YZ + a58YZ2 + a59YZW + a60YW + a61YW2 + a62Z + a63Z2 + a64Z3 + a65Z2W + a66ZW + a67ZW2 + a68W + a69W2 + a70W3
Z = a71 + a72X + a73X2 + a74X3 + a75X2Y + a76X2Z + a77X2W + a78XY + a79XY2 + a80XYZ + a81XYW + a82XZ + a83XZ2 + a84XZW + a85XW + a86XW2 + a87Y + a88Y2 + a89Y3 + a90Y2Z + a91Y2W + a92YZ + a93YZ2 + a94YZW + a95YW + a96YW2 + a97Z + a98Z2 + a99Z3 + a100Z2W + a101ZW + a102ZW2 + a103W + a104W2 + a105W3
W = a106 + a107X + a108X2 + a109X3 + a110X2Y + a111X2Z + a112X2W + a113XY + a114XY2 + a115XYZ + a116XYW + a117XZ + a118XZ2 + a119XZW + a120XW + a121XW2 + a122Y + a123Y2 + a124Y3 + a125Y2Z + a126Y2W + a127YZ + a128YZ2 + a129YZW + a130YW + a131YW2 + a132Z + a133Z2 + a134Z3 + a135Z2W + a136ZW + a137ZW2 + a138W + a139W2 + a140W3

Case O: D = 4 O = 4 M = 280

X = a1 + a2X + a3X2 + a4X3 + a5X4 + a6X3Y + a7X3Z + a8X3W + a9X2Y + a10X2Y2 + a11X2YZ + a12X2YW + a13X2Z + a14X2Z2 + a15X2ZW + a16X2W + a17X2W2 + a18XY + a19XY2 + a20XY3 + a21XY2Z + a22XY2W + a23XYZ + a24XYZ2 + a25XYZW + a26XYW + a27XYW2 + a28XZ + a29XZ2 + a30XZ3 + a31XZ2W + a32XZW + a33XZW2 + a34XW + a35XW2 + a36XW3 + a37Y + a38Y2 + a39Y3 + a40Y4 + a41Y3Z + a42Y3W + a43Y2Z + a44Y2Z2 + a45Y2ZW + a46Y2W + a47Y2W2 + a48YZ + a49YZ2 + a50YZ3 + a51YZ2W + a52YZW + a53YZW2 + a54YW + a55YW2 + a56YW3 + a57Z + a58Z2 + a59Z3 + a60Z4 + a61Z3W + a62Z2W + a63Z2W2 + a64ZW + a65ZW2 + a66ZW3 + a67W + a68W2 + a69W3 + a70W4

Y = a71 + a72X + a73X2 + a74X3 + a75X4 + a76X3Y + a77X3Z + a78X3W + a79X2Y + a80X2Y2 + a81X2YZ + a82X2YW + a83X2Z + a84X2Z2 + a85X2ZW + a86X2W + a87X2W2 + a88XY + a89XY2 + a90XY3 + a91XY2Z + a92XY2W + a93XYZ + a94XYZ2 + a95XYZW + a96XYW + a97XYW2 + a98XZ + a99XZ2 + a100XZ3 + a101XZ2W + a102XZW + a103XZW2 + a104XW + a105XW2 + a106XW3 + a107Y + a108Y2 + a109Y3 + a110Y4 + a111Y3Z + a112Y3W + a113Y2Z + a114Y2Z2 + a115Y2ZW + a116Y2W + a117Y2W2 + a118YZ + a119YZ2 + a120YZ3 + a121YZ2W + a122YZW + a123YZW2 + a124YW + a125YW2 + a126YW3 + a127Z + a128Z2 + a129Z3 + a130Z4 + a131Z3W + a132Z2W + a133Z2W2 + a134ZW + a135ZW2 + a136ZW3 + a137W + a138W2 + a139W3 + a140W4
Z = a141 + a142X + a143X2 + a144X3 + a145X4 + a146X3Y + a147X3Z + a148X3W + a149X2Y + a150X2Y2 + a151X2YZ + a152X2YW + a153X2Z + a154X2Z2 + a155X2ZW + a156X2W + a157X2W2 + a158XY + a159XY2 + a160XY3 + a161XY2Z + a162XY2W + a163XYZ + a164XYZ2 + a165XYZW + a166XYW + a167XYW2 + a168XZ + a169XZ2 + a170XZ3 + a171XZ2W + a172XZW + a173XZW2 + a174XW + a175XW2 + a176XW3 + a177Y + a178Y2 + a179Y3 + a180Y4 + a181Y3Z + a182Y3W + a183Y2Z + a184Y2Z2 + a185Y2ZW + a186Y2W + a187Y2W2 + a188YZ + a189YZ2 + a190YZ3 + a191YZ2W + a192YZW + a193YZW2 + a194YW + a195YW2 + a196YW3 + a197Z + a198Z2 + a199Z3 + a200Z4 + a201Z3W + a202Z2W + a203Z2W2 + a204ZW + a205ZW2 + a206ZW3 + a207W + a208W2 + a209W3 + a210W4
W = a211 + a212X + a213X2 + a214X3 + a215X4 + a216X3Y + a217X3Z + a218X3W + a219X2Y + a220X2Y2 + a221X2YZ + a222X2YW + a223X2Z + a224X2Z2 + a225X2ZW + a226X2W + a227X2W2 + a228XY + a229XY2 + a230XY3 + a231XY2Z + a232XY2W + a233XYZ + a234XYZ2 + a235XYZW + a236XYW + a237XYW2 + a238XZ + a239XZ2 + a240XZ3 + a241XZ2W + a242XZW + a243XZW2 + a244XW + a245XW2 + a246XW3 + a247Y + a248Y2 + a249Y3 + a250Y4 + a251Y3Z + a252Y3W + a253Y2Z + a254Y2Z2 + a255Y2ZW + a256Y2W + a257Y2W2 + a258YZ + a259YZ2 + a260YZ3 + a261YZ2W + a262YZW + a263YZW2 + a264YW + a265YW2 + a266YW3 + a267Z + a268Z2 + a269Z3 + a270Z4 + a271Z3W + a272Z2W + a273Z2W2 + a274ZW + a275ZW2 + a276ZW3 + a277W + a278W2 + a279W3 + a280W4