OneDimensional Maps
Chaos And TimeSeries Analysis
9/12/00 Lecture #2 in Physics 505

Dynamical systems

Random (stochastic) versus deterministic

Linear versus nonlinear

Simple (few variables) versus complex (many variables)

Examples (solar system, stock market, ecology, ...)

Some Properties of chaotic dynamical systems

Deterministic, nonlinear dynamics (necessary but not sufficient)

Aperiodic behavior (never repeats  infinite period)

Sensitive dependence on initial conditions (exponential)

Dependence on a control parameter (bifurcation, phase transition)

Perioddoubling route to chaos (common, but not universal)

Demonstrations

Computer animations (3body problem,
driven
pendulum)

Chaotic pendulums

Ball on oscillating floor

Falling leaf (or piece of paper)

Fluids (mixing, air hose, dripping faucet)

Chaotic water bucket

Chaotic electrical circuits
Logistic Equation  Motivation

Exhibits many aspects of chaotic systems (prototype)

Mathematically simple

Involves only a single variable

Doesn't require calculus

Exact solutions can be obtained

Can model many different phenomena

Ecology

Cancer growth

Finance

Etc...

Can be understood graphically
Exponential Growth (Discrete
Time)

X_{n}_{+1} = AX_{n}
(example: compound interest)

Example of linear deterministic dynamics

Example of an iterated map (involves feedback)

Exhibits stretching (A > 1) or shrinking (A
< 1)

Attracts to X = 0 (for A < 1) or X = infinity
(for A > 1)

Solution is X_{n} = X_{0}A^{n}
(exponential growth or decay)

A is the control parameter (the "knob")

A = 1 is a bifurcation point.
Logistic Equation

X_{n}_{+1} = AX_{n}(1  X_{n})

Quadratic nonlinearity (X^{2})

Graph of X_{n}_{+1}
versus X_{n} is a parabola

Equivalent form: Y_{n}_{+1} = B  Y_{n}^{2}
(quadratic map)

Y = A(X  0.5)

B = A^{2}/4  A/2

Solutions: X^{*} = 0, 1  1/A (fixed
point)

Graphical solution (reflection from
45° line  "cobweb diagram")

Computer simulation of logistic map

0 < A < 1 Case:

Only nonnegative solution is X^{*} = 0

All X_{0} in the interval 0 < X_{0} <
1 attract to X^{*}

They lie in the basin of attraction

The nonlinearity doesn't matter

1 < A < 3 Case:

Solution at X^{*} = 0 becomes a repellor

Solution at X^{*} = 1  1/A appears

It is a point attractor (also called "period1 cycle")

Basin of attraction is 0 < X_{0} < 1

3 < A < 3.449... Case:

Attractor at 1  1/A becomes unstable (repellor)

This happens when df/dX < 1 (==> A > 3)

This bifurcation is called a flip

Growing oscillation occurs

Oscillation nonlinearly saturates (period2 cycle)

X_{n}_{+2} = f(f(X_{n}))
= f^{(2)}(X_{n}) = X_{n}

Quartic equation has four roots

Two are the original unstable fixed points

The other two are are the new 2cycle

3.449... < A < 3.5699... Case:

Period2 becomes unstable when df^{(2)}(X)/dX
< 1

At this value (A = 3.440...) a stable period4 cycle is born

The process continues with successive period doublings

Infinite period is reached at A = 3.5699... (Feigenbaum point)

This is perioddoubling
route to chaos

Bifurcation plot is selfsimilar (a fractal)

Feigenvalues:
delta
= 4.6692..., alpha = 2.5029...

Feigenvalues are universal (for all smooth 1D unimodal maps)

3.5699... < A < 4 Case:

Most values of A in this range produce chaos (infinite period)

There are infinitely many periodic windows

Each periodic window displays period doubling

All periods are present somewhere for 3 < A < 4

A = 4 Case:

This value of A is special

It maps the interval 0 < X < 1 back onto itself (endomorphism)

Notice the fold at X_{n} = 0.5

Thus we have stretching and folding (silly putty demo)

Stretching is not uniform (cf: tent
map)

Each X_{n}_{+1} has two possible values of
X_{n}
(preimages)

Error in initial condition doubles
(on average) with each iteration

We lose 1 bit of precision with each time step

A > 4 Case:

Transient chaos for A slightly above 4 for most X_{0}

Orbit eventually escapes to infinity for most X_{0}
Other Properties
of the Logistic Map (A = 4)

Eventually fixed points

X_{0} = 0 and X_{0} = 1  1/A = 0.75
are (unstable) fixed points

X_{0} = 0.5 > 1 > 0 is an eventually fixed point

There are infinitely many such eventually fixed points

Each fixed point has two preimages, etc..., all eventually fixed

Although infinite in number they are a set of measure zero

They constitute a Cantor
set (Georg Cantor)

Compare with rational and irrational numbers

Eventually periodic points

If X_{n}_{+2} = X_{n} orbit is (unstable)
period2
cycle

Solution (A = 4): X^{*} = 0, 0.345491,
0.75, 0.904508

0 and 0.75 are (unstable) fixed points (as above)

0.345491 and 0.904508 are (unstable) period2 cycle

All periods are present and all are unstable

(Unstable) period3 orbit implies chaos (Li and Yorke)

Each period has infinitely many preimages

Still, most points are aperiodic (100%)

Periodic orbits are dense on the set

Probability density (also called invariant measure)

Many X_{n} values map to X_{n}_{+1}
close to 1.0

These in turn map to X_{n}_{+2} close to 0.0

Thus the probability density peaks at
0 and 1

Actual form: P = 1 / pi[X(1  X)]^{1/2}

Ergodic hypothesis: the average over all starting points is the
same as the average over time for a single starting point

Nonrecursive representation

X_{n} = (1  cos(2^{n}cos^{1}(1
 2X_{0})))/2

Ref: H. G. Schuster, Deterministic Chaos, (VCH, Weinheim,
1989)
Other OneDimensional
Maps

Sine map

X_{n}_{+1} = A sin(pi X_{n})

Properties similar to logistic map (except A = 1 corresponds
to A = 4)

Tent map

X_{n}_{+1} = A min(X_{n},
1  X_{n})

Piecewise linear

Uniform stretching

All orbits become unstable at
A = 1

Uniform (constant) probability density at A = 2

Numerical difficulties

General symmetric map

X_{n}_{+1} = A(1  2X_{n}
 1^{alpha})

alpha = 1 gives the tent map

alpha = 2 gives the logistic map

alpha is a measure of the smoothness of the map

Binary shift map

X_{n}_{+1} = 2X_{n} (mod 1)

Stretching, cutting, and reattaching

Resembles tent map

Chaotic only for irrational initial conditions

Can be used to generate pseudorandom numbers
J. C. Sprott  Physics 505
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