Dynamics of High-Dimensional Systems 11/10/2018 J. C. Sprott Department of Physics University of Wisconsin - Madison Presented at the Santa Fe Institute On July 27, 2004 Entire presentation available on WWW

Collaborators David Albers, SFI & U. Wisc - Physics Dee Dechert, U. Houston - Economics John Vano, U. Wisc - Math Joe Wildenberg, U. Wisc - Undergrad Jeff Noel, U. Wisc - Undergrad Mike Anderson, U. Wisc - Undergrad Sean Cornelius, U. Wisc - Undergrad Matt Sieth, U. Wisc - Undergrad

Typical Experimental Data 11/10/2018 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

How common is chaos? Logistic Map xn+1 = Axn(1 − xn) 1 11/10/2018 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) 11/10/2018 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 Iterated Quadratic Map 11/10/2018 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)

General 2-D Quadratic Maps 11/10/2018 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

High-Dimensional Quadratic Maps and Flows Extend to higher-degree polynomials...

Probability of Chaotic Solutions 11/10/2018 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

Correlation Dimension 11/10/2018 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 11/10/2018 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

Neural Net Architecture 11/10/2018 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 11/10/2018 % 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) D

Attractor Dimension N = 32 DKY = 0.46 D D

Routes to Chaos at Low D

Routes to Chaos at High D

Multispecies Lotka-Volterra Model 11/10/2018 Let xi be population of the ith species (rabbits, trees, people, stocks, …) dxi / dt = rixi (1 − Σ aijxj ) Parameters of the model: Vector of growth rates ri Matrix of interactions aij Number of species N N j=1

Parameters of the Model Growth rates Interaction matrix 1 r2 r3 r4 r5 r6 1 a12 a13 a14 a15 a16 a21 1 a23 a24 a25 a26 a31 a32 1 a34 a35 a36 a41 a42 a43 1 a45 a46 a51 a52 a53 a54 1 a56 a61 a62 a63 a64 a65 1

Choose ri and aij randomly from an exponential distribution: 1 P(a) = e−a P(a) a = − LOG(RND) mean = 1 a 5

Typical Time History 15 species xi Time

Probability of Chaos One case in 105 is chaotic for N = 4 with all species surviving Probability of coexisting chaos decreases with increasing N Evolution scheme: Decrease selected aij terms to prevent extinction Increase all aij terms to achieve chaos Evolve solutions at “edge of chaos” (small positive Lyapunov exponent)

Minimal High-D Chaotic L-V Model dxi /dt = xi(1 – xi-2 – xi – xi+1) 1

Space Time

Route to Chaos in Minimal LV Model

Other Simple High-D Models

Summary of High-D Dynamics 11/10/2018 Summary of High-D Dynamics Chaos is the rule Attractor dimension is ~ D/2 Lyapunov exponent tends to be small (“edge of chaos”) Quasiperiodic route is usual Systems are insensitive to parameter perturbations

11/10/2018 References http://sprott.physics.wisc.edu/ lectures/sfi2004.ppt (this talk) sprott@physics.wisc.edu (contact me)