Strange Attractors From Art to Science 11/16/2018 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 Entire presentation available on WWW

Outline Modeling of chaotic data Probability of chaos 11/16/2018 Outline Modeling of chaotic data Probability of chaos Examples of strange attractors Properties of strange attractors Attractor dimension scaling Lyapunov exponent scaling Aesthetics Simplest chaotic flows New chaotic electrical circuits

Typical Experimental Data 11/16/2018 Typical Experimental Data 5 x Not usually shown in textbooks Could be: Plasma fluctuations Stock market data Meteorological data EEG or EKG Ecological data etc... Until recently, no hope of detailed understanding Could be an example of deterministic chaos -5 Time 500

General 2-D Iterated Quadratic Map 11/16/2018 General 2-D Iterated Quadratic Map xn+1 = a1 + a2xn + a3xn2 + a4xnyn + a5yn + a6yn2 yn+1 = a7 + a8xn + a9xn2 + a10xnyn + a11yn + a12yn2 Equivalence (diffeomorphism) between systems with: 1 variable with many time delays Multiple variables and no delay Takens' theorem (1981)

Solutions Are Seldom Chaotic 11/16/2018 20 Chaotic Data (Lorenz equations) Chaotic Data (Lorenz equations) x Best attempt to fit previous equation to data from the Lorenz attractor Uses Principal Component Analysis (AKA: Singular Value Decomposition) Good short-term prediction (optimized for that) Bad long-term prediction (seldom chaotic) More complicated models don't help Solution of model equations Solution of model equations -20 Time 200

How common is chaos? Logistic Map xn+1 = Axn(1 - xn) 1 11/16/2018 1 Logistic Map xn+1 = Axn(1 - xn) Lyapunov Exponent Simplest 1-D chaotic system Chaotic over 13% of the range of A Solutions are unbounded outside the range plotted (hence unphysical) -1 -2 A 4

A 2-D Example (Hénon Map) 11/16/2018 2 b Simplest 2-D chaotic system Two control parameters Reduces to logistic map for b = 0 Chaotic "beach" on NW side of "island" occupies about 6% of area Does probability of chaos decrease with dimension? xn+1 = 1 + axn2 + bxn-1 -2 a -4 1

General 2-D Quadratic Map 11/16/2018 100 % Bounded solutions 10% Chaotic solutions 1% 12 coefficients ==> 12-D parameter space Coefficients chosen randomly in -amax < a < amax (hypercubes) 0.1% amax 0.1 1.0 10

Probability of Chaotic Solutions 11/16/2018 100% Iterated maps 10% Continuous flows (ODEs) 1% Quadratic maps and flows Chaotic flows must be at least 3-D Do the lines cross at high D? Is this result general or just for polynomials? 0.1% Dimension 1 10

Neural Net Architecture 11/16/2018 Neural Net Architecture Many architectures are possible Neural nets are universal approximators Output is bounded by squashing function Just another nonlinear map Can produce interesting dynamics tanh

% Chaotic in Neural Networks 11/16/2018 % Chaotic in Neural Networks Large collection of feedforward networks Single (hidden) layer (8 neurons, tanh squashing function) Randomly chosen weights (connection strengths with rms value s) d inputs (dimension / time lags)

Types of Attractors Examples: simple damped pendulum 11/16/2018 Types of Attractors Limit Cycle Fixed Point Spiral Radial Torus Strange Attractor Examples: simple damped pendulum driven mass on a spring inner tube

Strange Attractors Occur in infinite variety (like snowflakes) 11/16/2018 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 Occur in infinite variety (like snowflakes) Produced many millions, looked at over 100,000 Like pornography, know it when you see it

Stretching and Folding 11/16/2018 Stretching and Folding This is Poincare section for damped driven pendulum Taffy machine - Silly putty Shows sensitivity to initial conditions and fractal structure formation

Correlation Dimension 11/16/2018 Correlation Dimension 5 Correlation Dimension As described by Grassberger and Procaccia (1983) Error bars show standard deviation Scaling law approximately SQR(d) Useful experimental guidance 0.5 1 10 System Dimension

Lyapunov Exponent 10 1 0.1 0.01 1 10 Lyapunov Exponent 11/16/2018 Lyapunov Exponent 10 1 Lyapunov Exponent 0.1 Little difference in maps and flows At high-D chaos is more likely but weaker May relate to complex systems evolving “on the edge of chaos” 0.01 1 10 System Dimension

Aesthetic Evaluation 11/16/2018 7500 attractors of various Lyapunov exponents and Correlation Dimension Rated on scale of 1 - 5 by 3 artists and 4 scientists Strong preference for low L and intermediate F Some suggestion of individual preferences: - Scientists prefer simplicity - Artists more tolerant of complexity

“Simplest Dissipative Chaotic Flow” 11/16/2018 Sprott (1997) “Simplest Dissipative Chaotic Flow” dx/dt = y dy/dt = z dz/dt = -az + y2 - x 5 terms, 1 quadratic nonlinearity, 1 parameter

Linz and Sprott (1999) dx/dt = y dy/dt = z dz/dt = -az - y + |x| - 1 11/16/2018 Linz and Sprott (1999) dx/dt = y dy/dt = z dz/dt = -az - y + |x| - 1 6 terms, 1 abs nonlinearity, 2 parameters (but one =1)

11/16/2018 First Circuit

Bifurcation Diagram for First Circuit 11/16/2018 Bifurcation Diagram for First Circuit

11/16/2018 Second Circuit

11/16/2018 Third Circuit

11/16/2018 Chaos Circuit

Summary Chaos is the exception at low D Chaos is the rule at high D 11/16/2018 Summary Chaos is the exception at low D Chaos is the rule at high D Attractor dimension ~ D1/2 Lyapunov exponent decreases with increasing D New simple chaotic flows have been discovered New chaotic circuits have been developed