1 / 28

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

sheng
Download Presentation

Strange Attractors From Art to Science

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related