1. 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 at the
Santa Fe Institute
On June 20, 2000
Entire presentation available on WWW

2. Outline Modeling of chaotic data Probability of chaos
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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
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Typical Experimental Data 5 x
Not usually shown in textbooks
Could be:
Plasma fluctuations
Stock market data
Meteorological data
EEG or EKG
Ecological data
etc...
Until recently, no hope of detailed understanding
Could be an example of deterministic chaos -5
Time
500

4. General 2-D Iterated Quadratic Map
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General 2-D Iterated Quadratic Map
xn+1 = a1 + a2xn + a3xn2 + a4xnyn + a5yn + a6yn2
yn+1 = a7 + a8xn + a9xn2 + a10xnyn + a11yn + a12yn2
Equivalence (diffeomorphism) between systems with:
1 variable with many time delays
Multiple variables and no delay
Takens' theorem (1981)

5. Solutions Are Seldom Chaotic
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Chaotic Data (Lorenz equations)
Chaotic Data
(Lorenz equations) x
Best attempt to fit previous equation to data from the Lorenz attractor
Uses Principal Component Analysis
(AKA: Singular Value Decomposition)
Good short-term prediction (optimized for that)
Bad long-term prediction (seldom chaotic)
More complicated models don't help
Solution of model equations
Solution of model equations
-20
Time
200

6. How common is chaos? Logistic Map xn+1 = Axn(1 - xn) 1
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Logistic Map
xn+1 = Axn(1 - xn)
Lyapunov Exponent
Simplest 1-D chaotic system
Chaotic over 13% of the range of A
Solutions are unbounded outside the range plotted (hence unphysical) -1 -2 A 4

7. A 2-D Example (Hénon Map)
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Simplest 2-D chaotic system
Two control parameters
Reduces to logistic map for b = 0
Chaotic "beach" on NW side of "island" occupies about 6% of area
Does probability of chaos decrease with dimension?
xn+1 = 1 + axn2 + bxn-1 -2 a -4 1

8. General 2-D Quadratic Map
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100 %
Bounded solutions
10%
Chaotic solutions 1%
12 coefficients ==> 12-D parameter space
Coefficients chosen randomly in -amax < a < amax (hypercubes)
0.1%
amax
0.1
1.0 10

9. Probability of Chaotic Solutions
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100%
Iterated maps
10%
Continuous flows (ODEs) 1%
Quadratic maps and flows
Chaotic flows must be at least 3-D
Do the lines cross at high D?
Is this result general or just for polynomials?
0.1%
Dimension 1 10

10. Neural Net Architecture
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Neural Net Architecture
Many architectures are possible
Neural nets are universal approximators
Output is bounded by squashing function
Just another nonlinear map
Can produce interesting dynamics
tanh

11. % Chaotic in Neural Networks
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% Chaotic in Neural Networks
Large collection of feedforward networks
Single (hidden) layer (8 neurons, tanh squashing function)
Randomly chosen weights (connection strengths with rms value s)
d inputs (dimension / time lags)

12. Types of Attractors Examples: simple damped pendulum
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Types of Attractors
Limit Cycle
Fixed Point
Spiral
Radial
Torus
Strange Attractor
Examples:
simple damped pendulum
driven mass on a spring
inner tube

13. Strange Attractors Occur in infinite variety (like snowflakes)
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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
Occur in infinite variety (like snowflakes)
Produced many millions, looked at over 100,000
Like pornography, know it when you see it

14. Stretching and Folding
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Stretching and Folding
This is Poincare section for damped driven pendulum
Taffy machine - Silly putty
Shows sensitivity to initial conditions and fractal structure formation

15. Correlation Dimension
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Correlation Dimension 5
Correlation Dimension
As described by Grassberger and Procaccia (1983)
Error bars show standard deviation
Scaling law approximately SQR(d)
Useful experimental guidance
0.5 1 10
System Dimension

16. Lyapunov Exponent 10 1 0.1 0.01 1 10 Lyapunov Exponent
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Lyapunov Exponent 10 1
Lyapunov Exponent
0.1
Little difference in maps and flows
At high-D chaos is more likely but weaker
May relate to complex systems evolving “on the edge of chaos”
0.01 1 10
System Dimension

17. Aesthetic Evaluation
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7500 attractors of various Lyapunov exponents and Correlation Dimension
Rated on scale of by 3 artists and 4 scientists
Strong preference for low L and intermediate F
Some suggestion of individual preferences:
- Scientists prefer simplicity
- Artists more tolerant of complexity

18. “Simplest Dissipative Chaotic Flow”
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Sprott (1997)
“Simplest Dissipative Chaotic Flow”
dx/dt = y
dy/dt = z
dz/dt = -az + y2 - x
5 terms, 1 quadratic nonlinearity, 1 parameter

19. Linz and Sprott (1999) dx/dt = y dy/dt = z dz/dt = -az - y + |x| - 1
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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)

20. 11/16/2018
First Circuit

21. Bifurcation Diagram for First Circuit
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Bifurcation Diagram for First Circuit

22. 11/16/2018
Second Circuit

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Third Circuit

24. 11/16/2018
Chaos Circuit

25. Summary Chaos is the exception at low D Chaos is the rule at high D
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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