Strange Attractors From Art to Science

J. C. SprottDepartment of Physics

University of Wisconsin - Madison

Presented at the

Santa Fe Institute

On June 20, 2000

Outline Modeling of chaotic data Probability of chaos Examples of strange attractors Properties of strange attractors Attractor dimension scaling Lyapunov exponent scaling Aesthetics Simplest chaotic flows New chaotic electrical circuits

Typical Experimental Data

Time0 500

x

5

-5

General 2-D Iterated Quadratic 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

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

Aesthetic Evaluation

Sprott (1997)

dx/dt = y

dy/dt = z

dz/dt = -az + y2 - x

5 terms, 1 quadratic

nonlinearity, 1 parameter

“Simplest Dissipative Chaotic Flow”

xxxax 2

Linz and Sprott (1999)

dx/dt = y

dy/dt = z

dz/dt = -az - y + |x| - 1

6 terms, 1 abs nonlinearity, 2 parameters (but one =1)

1 xxxax

First Circuit

1 xxxax

Bifurcation Diagram for First Circuit

Second Circuit

Third Circuit

)sgn(xxxxax

Chaos Circuit

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

New chaotic circuits have been developed