Strange Attractors From Art to Science

J. C. SprottDepartment of PhysicsUniversity of Wisconsin - Madison

Presented to theSociety for chaos theory in psychology and the life sciencesOn August 1, 1997

Outline Modeling of chaotic data Probability of chaos Examples of strange attractors Properties of strange attractors Attractor dimension Simplest chaotic flow Chaotic surrogate models Aesthetics

Typical Experimental Data

Time0 500

x

5

-5

Determinism

xn+1 = f (xn, xn-1, xn-2, …)

where f is some model equation with adjustable parameters

Example (2-D Quadratic Iterated Map)

xn+1 = a1 + a2xn + a3xn2 +

a4xnyn + a5yn + a6yn2

yn+1 = a7 + a8xn + a9xn2 +

a10xnyn + a11yn + a12yn2

Solutions Are Seldom ChaoticChaotic Data (Lorenz equations)

Solution of model equations

Chaotic Data(Lorenz equations)

Solution of model equations

Time0 200

x

20

-20

How common is chaos?

Logistic Map

xn+1 = Axn(1 - xn)

-2 4A

Lyap

unov

Ex

pone

nt1

-1

A 2-D example (Hénon map)2

b

-2a-4 1

xn+1 = 1 + axn2 + bxn-1

Mandelbrot set

a

b

xn+1 = xn2 - yn

2 + a

yn+1 = 2xnyn + b

General 2-D quadratic map100 %

10%

1%

0.1%

Bounded solutions

Chaotic solutions

0.1 1.0 10amax

Probability of chaotic solutions

Iterated maps

Continuous flows (ODEs)

100%

10%

1%

0.1%1 10Dimension

% Chaotic in neural networks

Examples of strange attractors A collection of favorites New attractors generated in real ti

me Simplest chaotic flow Stretching and folding

Strange attractors Limit set as t Set of measure zero Basin of attraction Fractal structure

non-integer dimension self-similarity infinite detail

Chaotic dynamics sensitivity to initial conditions topological transitivity dense periodic orbits

Aesthetic appeal

Correlation dimension5

0.51 10System Dimension

Cor

rela

tion

Dim

ensi

on

Simplest chaotic flow

dx/dt = ydy/dt = zdz/dt = -x + y2 - Az 2.0168 < A < 2.0577

02 xxxAx

Chaotic surrogate modelsxn+1 = .671 - .416xn - 1.014xn

2 + 1.738xnxn-1 +.836xn-1 -.814xn-12

Data

Model

Auto-correlation function (1/f noise)

Aesthetic evaluation

References http://sprott.physics.wisc.edu/ lectu

res/satalk/ Strange Attractors: Creating Patter

ns in Chaos (M&T Books, 1993)

Chaos Demonstrations software Chaos Data Analyzer software sprott@juno.physics.wisc.edu