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 n Modeling of chaotic data n Probability of chaos n Examples of strange attractors n Properties of strange attractors n Attractor dimension n Simplest chaotic flow n Chaotic surrogate models n Aesthetics

Typical Experimental Data Time0 500 x 5 -5

Determinism x n+ 1 = f (x n, x n- 1, x n- 2, …) where f is some model equation with adjustable parameters

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

Solutions Are Seldom Chaotic Chaotic Data (Lorenz equations) Solution of model equations Chaotic Data (Lorenz equations) Solution of model equations Time0200 x

How common is chaos? Logistic Map x n +1 = Ax n (1 - x n ) -24A Lyapunov Exponent 1

A 2-D example (Hénon map) 2 b -2 a -41 x n +1 = 1 + ax n 2 + bx n -1

Mandelbrot set a b x n +1 = x n 2 - y n 2 + a y n +1 = 2x n y n + b

General 2-D quadratic map 100 % 10% 1% 0.1% Bounded solutions Chaotic solutions a max

Probability of chaotic solutions Iterated maps Continuous flows (ODEs) 100% 10% 1% 0.1% 110 Dimension

% Chaotic in neural networks

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

Strange attractors n Limit set as t   n Set of measure zero n Basin of attraction n Fractal structure u non-integer dimension u self-similarity u infinite detail n Chaotic dynamics u sensitivity to initial conditions u topological transitivity u dense periodic orbits n Aesthetic appeal

Correlation dimension System Dimension Correlation Dimension

Simplest chaotic flow dx/dt = y dy/dt = z dz/dt = -x + y 2 - Az < A <

Chaotic surrogate models x n+1 = x n x n x n x n x n x n-1 2 Data Model Auto-correlation function (1/f noise)

Aesthetic evaluation

References n lectures/satalk/ lectures/satalk/ n Strange Attractors: Creating Patterns in Chaos (M&T Books, 1993) Strange Attractors: Creating Patterns in Chaos n Chaos Demonstrations software Chaos Demonstrations n Chaos Data Analyzer software Chaos Data Analyzer n