1. Strange Attractors From Art to Science J. C. Sprott Department of Physics University of Wisconsin - Madison Presented at the Santa Fe Institute On June 20, 2000

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

5. 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

6. How common is chaos? Logistic Map x n +1 = Ax n (1 - x n ) -2 4 A Lyapunov Exponent 1 -1

7. A 2-D Example (Hénon Map) 2 b -2 a -4 1 x n +1 = 1 + ax n 2 + bx n -1

8. General 2-D Quadratic Map 100 % 10% 1% 0.1% Bounded solutions Chaotic solutions 0.1 1.0 10 a max

9. Probability of Chaotic Solutions Iterated maps Continuous flows (ODEs) 100% 10% 1% 0.1% 1 10 Dimension

10. Neural Net Architecture tanh

11. % Chaotic in Neural Networks

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

14. Stretching and Folding

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

16. Lyapunov Exponent 1 10 System Dimension Lyapunov Exponent 10 1 0.1 0.01

17. Aesthetic Evaluation

20. First Circuit

21. Bifurcation Diagram for First Circuit

22. Second Circuit

23. Third Circuit

24. Chaos Circuit