Strange Attractors From Art to Science

1
Strange 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

2
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

3
Typical Experimental Data
5
x
-5
500
Time
0
4
General 2-D Iterated Quadratic 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
6
How common is chaos?
1
Logistic Map xn1 Axn(1 - xn)
Lyapunov Exponent
-1
-2
4
A
7
A 2-D Example (Hénon Map)
2
b
xn1 1 axn2 bxn-1
-2
a
-4
1
8
General 2-D Quadratic Map
100
Bounded solutions
10
Chaotic solutions
1
0.1
amax
0.1
1.0
10
9
Probability of Chaotic Solutions
100
Iterated maps
10
Continuous flows (ODEs)
1
0.1
Dimension
1
10
10
Neural Net Architecture
tanh
11
Chaotic in Neural Networks
12
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
15
Correlation Dimension
5
Correlation Dimension
0.5
1
10
System Dimension
16
Lyapunov Exponent
10
1
Lyapunov Exponent
0.1
0.01
1
10
System Dimension
17
Aesthetic Evaluation
18
Sprott (1997)
Simplest Dissipative Chaotic Flow

• dx/dt y
• dy/dt z
• dz/dt -az y2 - x
• 5 terms, 1 quadratic nonlinearity, 1 parameter

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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)

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First Circuit
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Bifurcation Diagram for First Circuit
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Second Circuit
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Third Circuit
24
Chaos Circuit
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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
• New chaotic circuits have been developed