Gen AI in Business - Global Trends Report 2024.pdf
Strange Attractors
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
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