Sacolloq

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  • 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
  • Sacolloq

    1. 1. 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
    2. 2. Outline <ul><li>Modeling of chaotic data </li></ul><ul><li>Probability of chaos </li></ul><ul><li>Examples of strange attractors </li></ul><ul><li>Properties of strange attractors </li></ul><ul><li>Attractor dimension </li></ul><ul><li>Lyapunov exponent </li></ul><ul><li>Simplest chaotic flow </li></ul><ul><li>Chaotic surrogate models </li></ul><ul><li>Aesthetics </li></ul>
    3. 3. Acknowledgments <ul><li>Collaborators </li></ul><ul><ul><li>G. Rowlands (physics) U. Warwick </li></ul></ul><ul><ul><li>C. A. Pickover (biology) IBM Watson </li></ul></ul><ul><ul><li>W. D. Dechert (economics) U. Houston </li></ul></ul><ul><ul><li>D. J. Aks (psychology) UW-Whitewater </li></ul></ul><ul><li>Former Students </li></ul><ul><ul><li>C. Watts - Auburn Univ </li></ul></ul><ul><ul><li>D. E. Newman - ORNL </li></ul></ul><ul><ul><li>B. Meloon - Cornell Univ </li></ul></ul><ul><li>Current Students </li></ul><ul><ul><li>K. A. Mirus </li></ul></ul><ul><ul><li>D. J. Albers </li></ul></ul>
    4. 4. Typical Experimental Data Time 0 500 x 5 -5
    5. 5. Determinism <ul><li>x n +1 = f ( x n , x n -1 , x n -2 , …) </li></ul><ul><li>where f is some model equation with adjustable parameters </li></ul>
    6. 6. Example (2-D Quadratic Iterated Map) <ul><li>x n +1 = a 1 + a 2 x n + a 3 x n 2 + a 4 x n y n + a 5 y n + a 6 y n 2 </li></ul><ul><li>y n +1 = a 7 + a 8 x n + a 9 x n 2 + a 10 x n y n + a 11 y n + a 12 y n 2 </li></ul>
    7. 7. Solutions Are Seldom Chaotic Chaotic Data (Lorenz equations) Solution of model equations Chaotic Data (Lorenz equations) Solution of model equations Time 0 200 x 20 -20
    8. 8. How common is chaos? Logistic Map x n +1 = Ax n (1 - x n ) -2 4 A Lyapunov Exponent 1 -1
    9. 9. A 2-D Example (Hénon Map) 2 b -2 a -4 1 x n +1 = 1 + ax n 2 + bx n -1
    10. 10. The Hénon Attractor x n +1 = 1 - 1.4 x n 2 + 0.3 x n -1
    11. 11. Mandelbrot Set a b x n +1 = x n 2 - y n 2 + a y n +1 = 2 x n y n + b z n +1 = z n 2 + c
    12. 12. Mandelbrot Images
    13. 13. General 2-D Quadratic Map 100 % 10% 1% 0.1% Bounded solutions Chaotic solutions 0.1 1.0 10 a max
    14. 14. Probability of Chaotic Solutions Iterated maps Continuous flows (ODEs) 100% 10% 1% 0.1% 1 10 Dimension
    15. 15. Neural Net Architecture tanh
    16. 16. % Chaotic in Neural Networks
    17. 17. Types of Attractors Fixed Point Limit Cycle Torus Strange Attractor Spiral Radial
    18. 18. Strange Attractors <ul><li>Limit set as t   </li></ul><ul><li>Set of measure zero </li></ul><ul><li>Basin of attraction </li></ul><ul><li>Fractal structure </li></ul><ul><ul><li>non-integer dimension </li></ul></ul><ul><ul><li>self-similarity </li></ul></ul><ul><ul><li>infinite detail </li></ul></ul><ul><li>Chaotic dynamics </li></ul><ul><ul><li>sensitivity to initial conditions </li></ul></ul><ul><ul><li>topological transitivity </li></ul></ul><ul><ul><li>dense periodic orbits </li></ul></ul><ul><li>Aesthetic appeal </li></ul>
    19. 19. Stretching and Folding
    20. 20. Correlation Dimension 5 0.5 1 10 System Dimension Correlation Dimension
    21. 21. Lyapunov Exponent 1 10 System Dimension Lyapunov Exponent 10 1 0.1 0.01
    22. 22. Simplest Chaotic Flow d x /d t = y d y /d t = z d z /d t = - x + y 2 - Az 2.0168 < A < 2.0577
    23. 23. Simplest Chaotic Flow Attractor
    24. 24. Simplest Conservative Chaotic Flow x + x - x 2 = - 0.01 ... .
    25. 25. Chaotic Surrogate Models x n +1 = .671 - .416 x n - 1.014 x n 2 + 1.738 x n x n -1 +.836 x n -1 -.814 x n -1 2 Data Model Auto-correlation function (1/ f noise)
    26. 26. Aesthetic Evaluation
    27. 27. Summary <ul><li>Chaos is the exception at low D </li></ul><ul><li>Chaos is the rule at high D </li></ul><ul><li>Attractor dimension ~ D 1/2 </li></ul><ul><li>Lyapunov exponent decreases with increasing D </li></ul><ul><li>New simple chaotic flows have been discovered </li></ul><ul><li>Strange attractors are pretty </li></ul>
    28. 28. References <ul><li>http://sprott.physics.wisc.edu/ lectures/ sacolloq / </li></ul><ul><li>Strange Attractors: Creating Patterns in Chaos (M&T Books, 1993) </li></ul><ul><li>Chaos Demonstrations software </li></ul><ul><li>Chaos Data Analyzer software </li></ul><ul><li>[email_address] </li></ul>

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