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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 Society for chaos theory in psychology and the life sciences On August 1, 1997. Outline. Modeling of chaotic data Probability of chaos

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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 Society for chaos theory in psychology and the life sciences On August 1, 1997

  2. Outline • Modeling of chaotic data • Probability of chaos • Examples of strange attractors • Properties of strange attractors • Attractor dimension • Simplest chaotic flow • Chaotic surrogate models • Aesthetics

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

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

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

  6. 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

  7. 1 How common is chaos? Logistic Map xn+1 = Axn(1 - xn) Lyapunov Exponent -1 -2 A 4

  8. A 2-D example (Hénon map) 2 b xn+1 = 1 + axn2 + bxn-1 -2 a -4 1

  9. Mandelbrot set xn+1 = xn2 - yn2 + a yn+1 = 2xnyn + b a b

  10. General 2-D quadratic map 100 % Bounded solutions 10% Chaotic solutions 1% 0.1% amax 0.1 1.0 10

  11. Probability of chaotic solutions 100% Iterated maps 10% Continuous flows (ODEs) 1% 0.1% Dimension 1 10

  12. % Chaotic in neural networks

  13. Examples of strange attractors • A collection of favorites • New attractors generated in real time • Simplest chaotic flow • Stretching and folding

  14. 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

  15. Correlation dimension 5 Correlation Dimension 0.5 1 10 System Dimension

  16. Simplest chaotic flow dx/dt = y dy/dt = z dz/dt = -x + y2 - Az 2.0168 < A < 2.0577

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

  18. Aesthetic evaluation

  19. http://sprott.physics.wisc.edu/ lectures/satalk/ 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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