Strange Attractors From Art to Science

1
Strange Attractors From Art to Science

• J. C. Sprott
• Department of Physics
• University of Wisconsin - Madison
• Presented to the
• University of Wisconsin - Madison Physics
  Colloquium
• On November 14, 1997

2
Outline

• Modeling of chaotic data
• Probability of chaos
• Examples of strange attractors
• Properties of strange attractors
• Attractor dimension
• Lyapunov exponent
• Simplest chaotic flow
• Chaotic surrogate models
• Aesthetics

3
Acknowledgments

• Collaborators
• G. Rowlands (physics) U. Warwick
• C. A. Pickover (biology) IBM Watson
• W. D. Dechert (economics) U. Houston
• D. J. Aks (psychology) UW-Whitewater
• Former Students
• C. Watts - Auburn Univ
• D. E. Newman - ORNL
• B. Meloon - Cornell Univ
• Current Students
• K. A. Mirus
• D. J. Albers

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Typical Experimental Data
5
x
-5
500
Time
0
5
Determinism

• xn1 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)

• xn1 a1 a2xn a3xn2 a4xnyn a5yn a6yn2
• yn1 a7 a8xn a9xn2 a10xnyn a11yn
  a12yn2

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Solutions Are Seldom Chaotic
20
Chaotic Data (Lorenz equations)
Chaotic Data (Lorenz equations)
x
Solution of model equations
Solution of model equations
-20
Time
0
200
8
How common is chaos?
1
Logistic Map xn1 Axn(1 - xn)
Lyapunov Exponent
-1
-2
4
A
9
A 2-D Example (Hénon Map)
2
b
xn1 1 axn2 bxn-1
-2
a
-4
1
10
The Hénon Attractor
xn1 1 - 1.4xn2 0.3xn-1
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Mandelbrot Set
xn1 xn2 - yn2 a yn1 2xnyn b
a
zn1 zn2 c
b
12
Mandelbrot Images
13
General 2-D Quadratic Map
100
Bounded solutions
10
Chaotic solutions
1
0.1
amax
0.1
1.0
10
14
Probability of Chaotic Solutions
100
Iterated maps
10
Continuous flows (ODEs)
1
0.1
Dimension
1
10
15
Neural Net Architecture
tanh
16
Chaotic in Neural Networks
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Types of Attractors
Limit Cycle
Fixed Point
Spiral
Radial
Torus
Strange Attractor
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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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Stretching and Folding
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Correlation Dimension
5
Correlation Dimension
0.5
1
10
System Dimension
21
Lyapunov Exponent
10
1
Lyapunov Exponent
0.1
0.01
1
10
System Dimension
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Simplest Chaotic Flow
dx/dt y dy/dt z dz/dt -x y2 - Az
2.0168 lt A lt 2.0577
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Simplest Chaotic Flow Attractor
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Simplest Conservative Chaotic Flow
...
.
x x - x2 - 0.01
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Chaotic Surrogate Models
xn1 .671 - .416xn - 1.014xn2 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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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
• Strange attractors are pretty

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References

• http//sprott.physics.wisc.edu/
  lectures/sacolloq/
• Strange Attractors Creating Patterns in Chaos
  (MT Books, 1993)
• Chaos Demonstrations software
• Chaos Data Analyzer software
• sprott_at_juno.physics.wisc.edu