strange attractors

25
Attractors From Art to Science J. C. Sprott Department of Physics University of Wisconsin - Madison Presented at the Santa Fe Institute On June 20, 2000

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Page 1: Strange Attractors

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

Page 2: Strange Attractors

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

Page 3: Strange Attractors

Typical Experimental Data

Time0 500

x

5

-5

Page 4: Strange Attractors

General 2-D Iterated Quadratic Map

xn+1 = a1 + a2xn + a3xn2 + a4xnyn + a5yn + a6yn2

yn+1 = a7 + a8xn + a9xn2 + a10xnyn + a11yn + a12yn2

Page 5: Strange Attractors

Solutions Are Seldom ChaoticChaotic Data (Lorenz equations)

Solution of model equations

Chaotic Data(Lorenz equations)

Solution of model equations

Time0 200

x

20

-20

Page 6: Strange Attractors

How common is chaos?

Logistic Map

xn+1 = Axn(1 - xn)

-2 4A

Lya

puno

v

Exp

onen

t1

-1

Page 7: Strange Attractors

A 2-D Example (Hénon Map)2

b

-2a-4 1

xn+1 = 1 + axn2 + bxn-1

Page 8: Strange Attractors

General 2-D Quadratic Map100 %

10%

1%

0.1%

Bounded solutions

Chaotic solutions

0.1 1.0 10amax

Page 9: Strange Attractors

Probability of Chaotic Solutions

Iterated maps

Continuous flows (ODEs)

100%

10%

1%

0.1%1 10Dimension

Page 10: Strange Attractors

Neural Net Architecture

tanh

Page 11: Strange Attractors

% Chaotic in Neural Networks

Page 12: Strange Attractors

Types of AttractorsFixed Point Limit Cycle

Torus Strange Attractor

Spiral Radial

Page 13: Strange Attractors

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

Page 14: Strange Attractors

Stretching and Folding

Page 15: Strange Attractors

Correlation Dimension5

0.51 10System Dimension

Cor

rela

tion

Dim

ensi

on

Page 16: Strange Attractors

Lyapunov Exponent

1 10System Dimension

Lya

puno

v E

xpon

ent

10

1

0.1

0.01

Page 17: Strange Attractors

Aesthetic Evaluation

Page 18: Strange Attractors

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

Page 19: Strange Attractors

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

Page 20: Strange Attractors

First Circuit

1 xxxax

Page 21: Strange Attractors

Bifurcation Diagram for First Circuit

Page 22: Strange Attractors

Second Circuit

Page 23: Strange Attractors

Third Circuit

)sgn(xxxxax

Page 24: Strange Attractors

Chaos Circuit

Page 25: Strange Attractors

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