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Strange Attractors From Art to Science

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. Outline. Modeling of chaotic data Probability of chaos Examples of strange attractors

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Strange Attractors From Art to Science

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  1. Strange AttractorsFrom 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

  4. Typical Experimental Data 5 x -5 500 0 Time

  5. Determinism xn+1 = f (xn, xn-1, xn-2, …) where f is some model equation with adjustable parameters

  6. Example (2-D Quadratic Iterated Map) xn+1 = a1 + a2xn + a3xn2 + a4xnyn + a5yn + a6yn2 yn+1 = a7 + a8xn + a9xn2 + a10xnyn + a11yn + a12yn2

  7. Solutions Are Seldom Chaotic 20 Chaotic Data (Lorenz equations) Chaotic Data (Lorenz equations) x Solution of model equations Solution of model equations -20 0 Time 200

  8. 1 How common is chaos? Logistic Map xn+1 = Axn(1 - xn) Lyapunov Exponent -1 -2 A 4

  9. A 2-D Example (Hénon Map) 2 b xn+1 = 1 + axn2 + bxn-1 -2 a -4 1

  10. The Hénon Attractor xn+1 = 1 - 1.4xn2 + 0.3xn-1

  11. Mandelbrot Set xn+1 = xn2 - yn2 + a yn+1 = 2xnyn + b a zn+1 = 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

  17. Types of Attractors Limit Cycle Fixed Point Spiral Radial Torus Strange Attractor

  18. 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

  19. Stretching and Folding

  20. 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

  22. Simplest Chaotic Flow dx/dt = y dy/dt = z dz/dt = -x + y2 - Az 2.0168 < A < 2.0577

  23. Simplest Chaotic Flow Attractor

  24. Simplest Conservative Chaotic Flow ... . x+x-x2=- 0.01

  25. Chaotic Surrogate Models xn+1 = .671 - .416xn- 1.014xn2 + 1.738xnxn-1 +.836xn-1 -.814xn-12 Data Model Auto-correlation function (1/f noise)

  26. Aesthetic Evaluation

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

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