Dynamics of High-Dimensional Systems

J. C. SprottDepartment of Physics

University of Wisconsin - Madison

Presented at the

Santa Fe Institute

On July 27, 2004

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

Time0 500

x

5

-5

How common is chaos?

Logistic Map

xn+1 = Axn(1 − xn)

-2 4A

Lya

puno

v

Exp

onen

t1

-1

A 2-D Example (Hénon Map)2

b

−2a−4 1

xn+1 = 1 + axn2 + bxn-1

General 2-D Iterated Quadratic Map

xn+1 = a1 + a2xn + a3xn2 +

a4xnyn + a5yn + a6yn2

yn+1 = a7 + a8xn + a9xn2 +

a10xnyn + a11yn + a12yn2

General 2-D Quadratic Maps100 %

10%

1%

0.1%

Bounded solutions

Chaotic solutions

0.1 1.0 10amax

High-Dimensional Quadratic Maps and Flows

)1(1

)1(1

)( 0 tkxD

k jkajatD

j jxatix

D

k kxjkajaD

j jxadt

idx

110

Extend to higher-degree polynomials...

Probability of Chaotic Solutions

Iterated maps

Continuous flows (ODEs)

100%

10%

1%

0.1%1 10Dimension

Correlation Dimension5

0.51 10System Dimension

Cor

rela

tion

Dim

ensi

on

Lyapunov Exponent

1 10System Dimension

Lya

puno

v E

xpon

ent

10

1

0.1

0.01

Neural Net Architecture

tanh

11

1tanh

1

jnn XD

jijW

N

iiX

% Chaotic in Neural Networks

D

Attractor Dimension

DKY = 0.46 D

D

N = 32

Routes to Chaos at Low D

Routes to Chaos at High D

Multispecies Lotka-Volterra Model

j=1

N

• 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

Parameters of the Model

1r2

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

Growthrates Interaction matrix

Choose ri and aij randomly from an exponential distribution:

P(a)

a00 5

1 P(a) = e−a

a = − LOG(RND)

mean = 1

Typical Time History

Time

xi

15 species

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

1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0

0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1

1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0

0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0

0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0

0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1

1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1

dxi /dt = xi(1 – xi-2 – xi – xi+1)

Time

Spac

e

Route to Chaos in Minimal LV Model

jxN

jijaix

dtidx

1

tanh

Other Simple High-D Models

)1(1

tanh)(

tjxN

j ijatix

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

References

http://sprott.physics.wisc.edu/ l

ectures/sfi2004.ppt

(this talk)

sprott@physics.wisc.edu

(contact me)