Chaotic Dynamics on Large Networks

J. C. SprottDepartment of Physics

University of Wisconsin - Madison

Presented at the

Chaotic Modeling and Simulation

International Conference

in Chania, Crete, Greece

on June 3, 2008

What is a complex system? Complex ≠ complicated Not real and imaginary parts Not very well defined Contains many interacting parts Interactions are nonlinear Contains feedback loops (+ and -) Cause and effect intermingled Driven out of equilibrium Evolves in time (not static) Usually chaotic (perhaps weakly) Can self-organize, adapt, learn

A Physicist’s Neuron

jN

jjxax

1tanhout

Ninputs

tanh x

x

2 4

1

3

A General Model (artificial neural network)

N neurons

N

ijj

jijiii xaxbx1

tanh

“Universal approximator,” N ∞

Solutions are bounded

Examples of Networks

System Agents Interaction State Source

Brain Neurons Synapses Firing rate Metabolism

Food Web Species Feeding Population Sunlight

Financial Market

Traders Trans-actions

Wealth Money

Political System

Voters Information Party affiliation

The Press

Other examples: War, religion, epidemics, organizations, …

Political System

tanh x

x

Republican

Democrat

Informationfrom others

Political “state”

N

jjj xabxx

1

tanh

a1

a2

a3 aj = ±1/√N, 0

Voter

Types of Dynamics

1. Static

2. Periodic

3. ChaoticArguably the most “healthy”Especially if only weakly so

“Dead”

“Stuck in a rut”

Equilibrium

Limit Cycle (or Torus)

Strange Attractor

Route to Chaos at Large N (=317)

jj

ijii xabxdtdx

317

1tanh/

“Quasi-periodic route to chaos”

400 Random networksFully connected

Typical Signals for Typical Network

Average Signal from all NeuronsAll +1

All −1

N =b =

3171/4

Simulated Elections100% Democrat

100% Republican

N =b =

3171/4

Strange AttractorsN =b =

101/4

Competition vs. Cooperation

jj

ijii xabxdtdx

317

1tanh/

500 Random networksFully connected

b = 1/4

Competition

Cooperation

Bidirectionality

jj

ijii xabxdtdx

317

1tanh/

250 Random networksFully connected

b = 1/4

Opposition

Reciprocity

Connectivity

jj

ijii xabxdtdx

317

1tanh/

250 Random networksN = 317, b = 1/4

Dilute Fully connected

1%

Network Size

jj

ijii xabxdtdxN

1

tanh/

750 Random networksFully connected

b = 1/4

N = 317

What is the Smallest Chaotic Net? dx1/dt = – bx1 + tanh(x4 – x2)

dx2/dt = – bx2 + tanh(x1 + x4)

dx3/dt = – bx3 + tanh(x1 + x2 – x4)

dx4/dt = – bx4 + tanh(x3 – x2)

StrangeAttractor

2-torus

Circulant Networksdxi /dt = −bxi + Σ ajxi+j

Fully Connected Circulant Network

jij

jii xabxdtdxN

1

1tanh/

N = 317

Diluted Circulant Network

)tanh(/ 25412642 iiiii xxxbxdtdx

N = 317

Near-Neighbor Circulant Network)tanh(/ 654321 iiiiiiii xxxxxxbxdtdx

N = 317

Summary of High-N Dynamics Chaos is generic for sufficiently-connected networks

Sparse, circulant networks can also be chaotic (but

the parameters must be carefully tuned)

Quasiperiodic route to chaos is usual

Symmetry-breaking, self-organization, pattern

formation, and spatio-temporal chaos occur

Maximum attractor dimension is of order N/2

Attractor is sensitive to parameter perturbations, but

dynamics are not

References

A paper on this topic is scheduled to

appear soon in the journal Chaos

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

networks.ppt (this talk)

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

(my chaos textbook)

sprott@physics.wisc.edu (contact me)