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Sacolloq
- 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. 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 Time 0 500 x 5 -5
- 5. Determinism x n +1 = f ( x n , x n -1 , x n -2 , …) where f is some model equation with adjustable parameters
- 6. Example (2-D Quadratic Iterated Map) 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 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
- 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. How common is chaos? Logistic Map x n +1 = Ax n (1 - x n ) -2 4 A Lyapunov Exponent 1 -1
- 9. A 2-D Example (Hénon Map) 2 b -2 a -4 1 x n +1 = 1 + ax n 2 + bx n -1
- 10. The Hénon Attractor x n +1 = 1 - 1.4 x n 2 + 0.3 x n -1
- 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
- 13. General 2-D Quadratic Map 100 % 10% 1% 0.1% Bounded solutions Chaotic solutions 0.1 1.0 10 a max
- 14. Probability of Chaotic Solutions Iterated maps Continuous flows (ODEs) 100% 10% 1% 0.1% 1 10 Dimension
- 15. Neural Net Architecture tanh
- 16. % Chaotic in Neural Networks
- 17. Types of Attractors Fixed Point Limit Cycle Torus Strange Attractor Spiral Radial
- 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
- 20. Correlation Dimension 5 0.5 1 10 System Dimension Correlation Dimension
- 21. Lyapunov Exponent 1 10 System Dimension Lyapunov Exponent 10 1 0.1 0.01
- 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. Simplest Chaotic Flow Attractor
- 24. Simplest Conservative Chaotic Flow x + x - x 2 = - 0.01 ... .
- 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)
- 27. Summary Chaos is the exception at low D Chaos is the rule at high D Attractor dimension ~ D 1/2 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 / Strange Attractors: Creating Patterns in Chaos (M&T Books, 1993) Chaos Demonstrations software Chaos Data Analyzer software [email_address]
Editor's Notes
- #2: 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