Strange Attractors From Art to Science

J. C. SprottDepartment of Physics

University of Wisconsin - Madison

Presented to the

University of Wisconsin - Madison Physics Colloquium

On November 14, 1997

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

Acknowledgments Collaborators

G. Rowlands (physics) U. Warwick C. A. Pickover (biology) IBM Watson W. D. Dechert (economics) U. Houston D. J. Aks (psychology) UW-Whitewater

Former Students C. Watts - Auburn Univ D. E. Newman - ORNL B. Meloon - Cornell Univ

Current Students K. A. Mirus D. J. Albers

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

Lya

puno

v

Exp

onen

t1

-1

A 2-D Example (Hénon Map)2

b

-2a-4 1

xn+1 = 1 + axn2 + bxn-1

The Hénon Attractorxn+1 = 1 - 1.4xn

2 + 0.3xn-1

Mandelbrot Set

a

b

xn+1 = xn2 - yn

2 + a

yn+1 = 2xnyn + b

zn+1 = zn2 + c

Mandelbrot Images

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

Neural Net Architecture

tanh

% Chaotic in Neural Networks

Types of AttractorsFixed Point Limit Cycle

Torus Strange Attractor

Spiral Radial

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

Stretching and Folding

Correlation Dimension5

0.51 10System Dimension

Cor

rela

tion

Dim

ensi

on

Lyapunov Exponent

1 10System Dimension

Lya

puno

v E

xpon

ent

10

1

0.1

0.01

Simplest Chaotic Flow

dx/dt = ydy/dt = zdz/dt = -x + y2 - Az

2.0168 < A < 2.0577

02 xxxAx

Simplest Chaotic Flow Attractor

Simplest Conservative Chaotic Flow

x + x - x2 = - 0.01... .

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

Summary Chaos is the exception at low D

Chaos is the rule at high D

Attractor dimension ~ D1/2

Lyapunov exponent decreases

with increasing D

New simple chaotic flows have

been discovered

Strange attractors are pretty

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

lectures/sacolloq/ Strange Attractors: Creating Pat

terns in Chaos (M&T Books, 1993)

Chaos Demonstrations software

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