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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

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

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Acknowledgments n Collaborators u G. Rowlands (physics) U. Warwick u C. A. Pickover (biology) IBM Watson u W. D. Dechert (economics) U. Houston u D. J. Aks (psychology) UW-Whitewater n Former Students u C. Watts - Auburn Univ u D. E. Newman - ORNL u B. Meloon - Cornell Univ n Current Students u K. A. Mirus u D. J. Albers

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Typical Experimental Data Time0 500 x 5 -5

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

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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

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

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

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

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The Hénon Attractor x n +1 = x n x n -1

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Mandelbrot Set a b x n +1 = x n 2 - y n 2 + a y n +1 = 2x n y n + b z n +1 = z n 2 + c

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Mandelbrot Images

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

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

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Neural Net Architecture tanh

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% Chaotic in Neural Networks

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Types of Attractors Fixed Point Limit Cycle TorusStrange Attractor SpiralRadial

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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

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Stretching and Folding

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Correlation Dimension System Dimension Correlation Dimension

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Lyapunov Exponent 110 System Dimension Lyapunov Exponent

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

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Simplest Chaotic Flow Attractor

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Simplest Conservative Chaotic Flow x + x - x 2 =

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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)

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Aesthetic Evaluation

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Summary n Chaos is the exception at low D n Chaos is the rule at high D n Attractor dimension ~ D 1/2 n Lyapunov exponent decreases with increasing D n New simple chaotic flows have been discovered n Strange attractors are pretty

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References n lectures/sacolloq/ lectures/sacolloq/ 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

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