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 to the
University of Wisconsin - Madison Physics Colloquium
On November 14, 1997
Keynote address at meeting of Society for Chaos Theory in Psychology and the Life Sciences last summer
New technology - PowerPoint
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
Lyapunov exponent
Simplest chaotic flow
Chaotic surrogate models
Aesthetics

3. 4/15/2017
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
Many people have been involved in various aspects of this work
These are coauthors of papers
Many fields are represented

4. 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

5. Determinism xn+1 = f (xn, xn-1, xn-2, …)
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Determinism
xn+1 = f (xn, xn-1, xn-2, …)
where f is some model equation with adjustable parameters
This is a discrete time dynamical system
Also called an iterated map
Can also have a continuous time flow (derivatives)

6. Example (2-D Quadratic Iterated Map)
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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
Equivalence (diffeomorphism) between systems with:
1 variable with many time delays
Multiple variables and no delay
Takens' theorem (1981)

7. 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

8. 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

9. 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

10. The Hénon Attractor xn+1 = 1 - 1.4xn2 + 0.3xn-1
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The Hénon Attractor
xn+1 = xn xn-1
Plot of next value of x versus present value
It is a strange attractor with fractal structure

11. Mandelbrot Set zn+1 = zn2 + c xn+1 = xn2 - yn2 + a yn+1 = 2xnyn + b a
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xn+1 = xn2 - yn2 + a
yn+1 = 2xnyn + b a
"The most complicated mathematical object ever seen"
Only the bounded orbits are shown
Boundary is fractal with dimension 2
All periods are represented somewhere
Chaotic orbits are very rare (probability zero?)
zn+1 = zn2 + c b

12. 4/15/2017
Mandelbrot Images
These are deep zooms into regions near the basin boundary
Colors indicate number of iterations required for an orbit starting at x = y = 0 to cross a circle of radius 2 (unbounded)

13. 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

14. 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

15. 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

16. % 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)

17. Types of Attractors Limit Cycle Fixed Point Torus Strange Attractor
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Types of Attractors
Limit Cycle
Fixed Point
Spiral
Radial
Torus
Strange Attractor
Demonstrate damped pendulum
Demonstrate driven mass on a spring
Demonstrate inner tube

18. 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

19. 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

20. 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

21. 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

22. dx/dt = y dy/dt = z dz/dt = -x + y2 - Az Simplest Chaotic Flow
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Simplest Chaotic Flow
dx/dt = y
dy/dt = z
dz/dt = -x + y2 - Az
< A <
Discovered last year
Jerk function: j = da/dt
(cf: jounce, sprite, surge, snap)
Compare
- Lorenz equations (7 variables, 2 nonlinearities)
- Rossler equations (7 variables, 1 nonlinearity)

23. Simplest Chaotic Flow Attractor
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Simplest Chaotic Flow Attractor
Attractor
- Dimension
- Approximately a Mobius strip
- Basin resembles a hairy tadpole

24. Simplest Conservative Chaotic Flow
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Simplest Conservative Chaotic Flow
Conservative (vs. dissipative) case
Liouville’s theorem is satisfied (phase-space volume is conserved) - time reversable
Shown is a Poincare section with a = 0 for many starting points - no attractor
Exhibits transient chaos over much of plane
Chaotic orbits near island chains surrounded by toroidal KAM surface
... .
x + x - x2 =

25. Chaotic Surrogate Models
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Chaotic Surrogate Models
xn+1 = xn xn xnxn xn xn-12
Data
Model
Can use arbitrary nonlinear functions
Brute-force method (variant of simulated annealing)
Easy to fit particular characteristics (i.e., power spectrum)
Not good for prediction (doesn't preserve dynamics)
Model is (usually) NOT unique, but (usually) IS robust to small variations of parameters
Auto-correlation function (1/f noise)

26. 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

27. 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
Strange attractors are pretty

28. References http://sprott.physics.wisc.edu/ lectures/sacolloq/
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References
lectures/sacolloq/
Strange Attractors: Creating Patterns in Chaos (M&T Books, 1993)
Chaos Demonstrations software
Chaos Data Analyzer software