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**Strange Attractors From Art to Science**

4/15/2017 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 Keynote address at meeting of Society for Chaos Theory in Psychology and the Life Sciences last summer New technology - PowerPoint Entire presentation available on WWW

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**Outline Modeling of chaotic data Probability of chaos**

4/15/2017 Outline Modeling of chaotic data Probability of chaos Examples of strange attractors Properties of strange attractors Attractor dimension Lyapunov exponent Simplest chaotic flow Chaotic surrogate models Aesthetics

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4/15/2017 Acknowledgments Collaborators G. Rowlands (physics) U. Warwick C. A. Pickover (biology) IBM Watson W. D. Dechert (economics) U. Houston D. J. Aks (psychology) UW-Whitewater Former Students C. Watts - Auburn Univ D. E. Newman - ORNL B. Meloon - Cornell Univ Current Students K. A. Mirus D. J. Albers Many people have been involved in various aspects of this work These are coauthors of papers Many fields are represented

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**Typical Experimental Data**

4/15/2017 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

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**Determinism xn+1 = f (xn, xn-1, xn-2, …)**

4/15/2017 Determinism xn+1 = f (xn, xn-1, xn-2, …) where f is some model equation with adjustable parameters This is a discrete time dynamical system Also called an iterated map Can also have a continuous time flow (derivatives)

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**Example (2-D Quadratic Iterated Map)**

4/15/2017 Example (2-D Quadratic Iterated 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)

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**Solutions Are Seldom Chaotic**

4/15/2017 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

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**How common is chaos? Logistic Map xn+1 = Axn(1 - xn) 1**

4/15/2017 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

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**A 2-D Example (Hénon Map)**

4/15/2017 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

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**The Hénon Attractor xn+1 = 1 - 1.4xn2 + 0.3xn-1**

4/15/2017 The Hénon Attractor xn+1 = xn xn-1 Plot of next value of x versus present value It is a strange attractor with fractal structure

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**Mandelbrot Set zn+1 = zn2 + c xn+1 = xn2 - yn2 + a yn+1 = 2xnyn + b a**

4/15/2017 xn+1 = xn2 - yn2 + a yn+1 = 2xnyn + b a "The most complicated mathematical object ever seen" Only the bounded orbits are shown Boundary is fractal with dimension 2 All periods are represented somewhere Chaotic orbits are very rare (probability zero?) zn+1 = zn2 + c b

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4/15/2017 Mandelbrot Images These are deep zooms into regions near the basin boundary Colors indicate number of iterations required for an orbit starting at x = y = 0 to cross a circle of radius 2 (unbounded)

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**General 2-D Quadratic Map**

4/15/2017 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

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**Probability of Chaotic Solutions**

4/15/2017 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

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**Neural Net Architecture**

4/15/2017 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

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**% Chaotic in Neural Networks**

4/15/2017 % 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)

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**Types of Attractors Limit Cycle Fixed Point Torus Strange Attractor**

4/15/2017 Types of Attractors Limit Cycle Fixed Point Spiral Radial Torus Strange Attractor Demonstrate damped pendulum Demonstrate driven mass on a spring Demonstrate inner tube

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**Strange Attractors Occur in infinite variety (like snowflakes)**

4/15/2017 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

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**Stretching and Folding**

4/15/2017 Stretching and Folding This is Poincare section for damped driven pendulum Taffy machine - Silly putty Shows sensitivity to initial conditions and fractal structure formation

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**Correlation Dimension**

4/15/2017 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

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**Lyapunov Exponent 10 1 0.1 0.01 1 10 Lyapunov Exponent**

4/15/2017 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

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**dx/dt = y dy/dt = z dz/dt = -x + y2 - Az Simplest Chaotic Flow**

4/15/2017 Simplest Chaotic Flow dx/dt = y dy/dt = z dz/dt = -x + y2 - Az < A < Discovered last year Jerk function: j = da/dt (cf: jounce, sprite, surge, snap) Compare - Lorenz equations (7 variables, 2 nonlinearities) - Rossler equations (7 variables, 1 nonlinearity)

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**Simplest Chaotic Flow Attractor**

4/15/2017 Simplest Chaotic Flow Attractor Attractor - Dimension - Approximately a Mobius strip - Basin resembles a hairy tadpole

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**Simplest Conservative Chaotic Flow**

4/15/2017 Simplest Conservative Chaotic Flow Conservative (vs. dissipative) case Liouville’s theorem is satisfied (phase-space volume is conserved) - time reversable Shown is a Poincare section with a = 0 for many starting points - no attractor Exhibits transient chaos over much of plane Chaotic orbits near island chains surrounded by toroidal KAM surface ... . x + x - x2 =

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**Chaotic Surrogate Models**

4/15/2017 Chaotic Surrogate Models xn+1 = xn xn xnxn xn xn-12 Data Model Can use arbitrary nonlinear functions Brute-force method (variant of simulated annealing) Easy to fit particular characteristics (i.e., power spectrum) Not good for prediction (doesn't preserve dynamics) Model is (usually) NOT unique, but (usually) IS robust to small variations of parameters Auto-correlation function (1/f noise)

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Aesthetic Evaluation 4/15/2017 7500 attractors of various Lyapunov exponents and Correlation Dimension Rated on scale of 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

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**Summary Chaos is the exception at low D Chaos is the rule at high D**

4/15/2017 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 Strange attractors are pretty

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**References http://sprott.physics.wisc.edu/ lectures/sacolloq/**

4/15/2017 References lectures/sacolloq/ Strange Attractors: Creating Patterns in Chaos (M&T Books, 1993) Chaos Demonstrations software Chaos Data Analyzer software

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