# Spatiotemporal Chaos and Complexity

## 12/12/00 Lecture #15 in Physics 505

### Announcements

• Some people have already earned an A (>25 points)
• You may wish to omit the last assignment(s)
• Homework #15 is due at 3:30 pm on 12/19 in my office or mailbox
• Note that HW #15 is graded differently from the others
• All late assignments are due by 3:30 pm on 12/19 in my office or mailbox
• Please fill out and return the teaching evaluation
• What topics did you enjoy most and least?
• Would you have preferred more depth and less breadth?
• How useful was it to have the lecture notes on the WWW?
• Was the one long lecture per week the best format for the course?
• Was the emphasis on computer experimentation appropriate?

### Comments on Homework #13 (Iterated Function Systems)

• Everyone had good plots of Sierpinski triangle and fern
• Should use small dots and many iterations
• Use non-uniform probabilities  (P ~ |det J| = |ad - bc|)
• IFS patterns are attractors and fractals, but not usually called "strange attractors"
• Bounded, linear IFS's are usually contracting in every direction, hence not chaotic
• Only one person calculated capacity dimension (1.585)

### Review (last week) - Non-Attracting Chaotic Sets

• Multifractals
• Fractals that are not uniformly covered
• Spectrum of generalized dimensions, Dq
• And generalized entropies, Kq
• Where -infinity < q < +infinity
• Summary of Time-Series Analysis
• Time-Series Analysis Tutorial (using CDA)
• Iterated Function Systems
• Multiple affine transformations
• Random iteration algorithm
• Collage theorem
• Image compression
• IFS clumpiness test

### Mandelbrot and Julia Sets

• Non-Attracting Chaotic Sets
• These sets ARE attracting
• They are generally only transiently chaotic
• Derivation from logistic equation
• Define a new variableZ = A(1/2 - X)
• Solve for X(Z, A) to get:  X = 1/2 - Z/A
• Substitute into logistic equation:  Zn+1 = Zn2 + c
• Where  c = A/2 - A2/4
• Range (1 < A < 4)  ==>  -2 < c < 1/2
• Zn+1 = Zn2 + c is equivalent to logistic map
• General the above to complex values of Z and c
• Review of complex numbers
• Z = X + iY, where i = (-1)1/2
• Z2 = X2 + 2iXY - Y2
• Separate real and imaginary parts
• Xn+1 = Xn2 - Yn2 + a
• Yn+1 = 2XnYn + b
• where a = Re(c) and b = Im(c)
• This is just another 2-D quadratic map
• X, Y, a, and b are real variables
• Orbits are either bounded or unbounded
• Mandelbrot (M) set
• Region of a-b space with bounded orbits with X0 = Y0 = 0
• Orbit escapes to infinity if X2 + Y2 > 4 (circle of radius 2)
• It's sometimes defined as the complement of this
• There is only one Mandelbrot set
• The "buds" in the M-set correspond to different periodicities
• Usually plotted are escape-time contours in colors
• Each point in the M-set has a corresponding Julia set
• The M-set is everywhere connected
• Boundary of M-set is fractal with dimension = 2 (proved)
• Area of set is ~ p/2
• Points along the real axis replicate logistic map and exhibit chaos
• Points just outside the boundary exhibit transient chaos
• The chaotic region appears to be a set of measure zero (not proved)
• Boundary of M-set is a repellor
• With deep zoom, M-set and J-set are identical
• People have zoomed in by factors as large as 101600
• Miniature M-sets are found at deep zooms
• See the Mandelbrot Java applet written by Andrew R. Cavender
• Julia (J) sets
• Region of X0-Y0 space with bounded orbits for given a, b
• Orbit escapes to infinity if X2 + Y2 > 4 (circle of radius 2)
• This is sometimes called the "filled-in" Julia set
• There are infinitely many J-sets
• Usually plotted are escape-time contours in colors
• The J-sets correspond to points on the Mandelbrot set
• J-sets from inside the M-set are connected
• J-sets from outside the M-set are "dusts"
• Boundary of J-set is a repellor
• With deep zoom, J-set and M-set are identical
• Fixed points of Julia sets
• Z = Z2 + c  ==>  Z = 1/2 ± (1 - 4c)1/2/2
• These fixed points are unstable (repellors)
• They can be found by backward iterationZn = ± (Zn+1 - c)1/2
• There are two roots (pre-images) each with two roots, etc.
• Find them with the random iteration algorithm (cf: IFS)
• The repelling boundary of J-set thus becomes an attractor
• An example is the Julia dendrite  (c = i)
• Generalized Julia sets
• Applications of M-set and J-sets
• None known except computer art

### Spatiotemporal Chaos (Complexity)

• Examples of spatiotemporal (infinite-dimensional) dynamics:
• Turbulent fluids
• Plasmas (ionized gases)
• The weather
• Molecular diffusion
• The brain
• Any process governed by partial differential equations (PDEs)
• Two coupled logistic maps
• Xn+1 = (1 - e) A1Xn (1 - Xn) + eA2 Yn (1 - Yn)

• Yn+1 = (1 - e) A2Yn (1 - Yn) + eA1 Xn (1 - Xn)
• Each map has a different A but the same coupling 0 < e < 1
• If one map is periodic and one chaotic, which wins?
• Can do the same with other maps and flows (Lorenz, etc.)
• Coupled-map lattices (CMLs)
• Consider a 1-D lattice of logistic maps
• Coupling can be to all others (GCMs) or N neighbors
• Dynamics exhibit transient chaos, waves, diffusion, damping, etc.
• Such systems have not been extensively studied
• Could use a distribution of e values
• Could extend calculations to 2 or more dimensions
• Cellular automata
• Langton's ants
• Example of simple computer automaton
• Lots of ways to extend these models
• Summary of spatiotemporal (complex) models
• Models can be discrete or continuous in space, time, and value:
•  space D, time D, value D cellular automata space D, time D, value C coupled map lattices space D, time C, value D not studied space D, time C, value C coupled flow lattices space C, time D, value D not studied space C, time D, value C not studied space C, time C, value D not studied space C, time C, value C partial differential eqns

### Concluding Remarks

• Nature is nonlinear and often chaotic
• Nature is complex, but simple models may suffice
• These models preclude prediction but invite control
• Remember the butterfly!