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
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
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
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
Lyap
unov
Ex
pone
nt1
-1
A 2-D example (Hénon map)2
b
-2a-4 1
xn+1 = 1 + axn2 + bxn-1
Mandelbrot set
a
b
xn+1 = xn2 - yn
2 + a
yn+1 = 2xnyn + b
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
% Chaotic in neural networks
Examples of strange attractors A collection of favorites New attractors generated in real ti
me Simplest chaotic flow Stretching and folding
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
Correlation dimension5
0.51 10System Dimension
Cor
rela
tion
Dim
ensi
on
Simplest chaotic flow
dx/dt = ydy/dt = zdz/dt = -x + y2 - Az 2.0168 < A < 2.0577
02 xxxAx
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
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]