strange attractors from art to science
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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 PresentationTRANSCRIPT
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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]