Strange Attractors From Art to Science

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Attractors From Art to Science J. C. Sprott Department of Physics University of Wisconsin - Madison Presented to the Society for chaos theor y in psychology and the life sciences On August 1, 1997

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Strange Attractors From Art to Science. J. C. Sprott Department of Physics University of Wisconsin - Madison Presented to the Society for chaos theory in psychology and the life sciences On August 1, 1997. Outline. Modeling of chaotic data Probability of chaos - PowerPoint PPT Presentation

Transcript of Strange Attractors From Art to Science

Page 1: Strange Attractors From Art to Science

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

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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

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Typical Experimental Data

Time0 500

x

5

-5

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Determinism

xn+1 = f (xn, xn-1, xn-2, …)

where f is some model equation with adjustable parameters

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Example (2-D Quadratic Iterated Map)

xn+1 = a1 + a2xn + a3xn2 +

a4xnyn + a5yn + a6yn2

yn+1 = a7 + a8xn + a9xn2 +

a10xnyn + a11yn + a12yn2

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Solutions Are Seldom ChaoticChaotic Data (Lorenz equations)

Solution of model equations

Chaotic Data(Lorenz equations)

Solution of model equations

Time0 200

x

20

-20

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How common is chaos?

Logistic Map

xn+1 = Axn(1 - xn)

-2 4A

Lyap

unov

Ex

pone

nt1

-1

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A 2-D example (Hénon map)2

b

-2a-4 1

xn+1 = 1 + axn2 + bxn-1

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Mandelbrot set

a

b

xn+1 = xn2 - yn

2 + a

yn+1 = 2xnyn + b

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General 2-D quadratic map100 %

10%

1%

0.1%

Bounded solutions

Chaotic solutions

0.1 1.0 10amax

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Probability of chaotic solutions

Iterated maps

Continuous flows (ODEs)

100%

10%

1%

0.1%1 10Dimension

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% Chaotic in neural networks

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

me Simplest chaotic flow Stretching and folding

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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

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Correlation dimension5

0.51 10System Dimension

Cor

rela

tion

Dim

ensi

on

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Simplest chaotic flow

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

02 xxxAx

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

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

Data

Model

Auto-correlation function (1/f noise)

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Aesthetic evaluation

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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 [email protected]