Chaotic Dynamics on Large Networks J. C. Sprott Department of Physics University of Wisconsin - Madison Presented at the Chaotic Modeling and Simulation International Conference in Chania, Crete, Greece on June 3, 2008

Collaborators n David Albers n Sean Cornelius

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

A Physicist’s Neuron N inputs tanh x x

A General Model (artificial neural network) N neurons “Universal approximator,” N  ∞ Solutions are bounded

Examples of Networks SystemAgentsInteractionStateSource BrainNeuronsSynapsesFiring rateMetabolism Food WebSpeciesFeedingPopulationSunlight Financial Market TradersTrans- actions WealthMoney Political System VotersInformationParty affiliation The Press Other examples: War, religion, epidemics, organizations, …

Political System tanh x x Republican Democrat Information from others Political “state” a1a1 a2a2 a3a3 a j = ±1/√N, 0 Voter

Types of Dynamics 1. Static 2. Periodic 3. Chaotic Arguably 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) “Quasi-periodic route to chaos” 400 Random networks Fully connected

Typical Signals for Typical Network

Average Signal from all Neurons All +1 All −1 N = b = 317 1/4

Simulated Elections 100% Democrat 100% Republican N = b = 317 1/4

Real Electroencephlagrams

Strange Attractors N = b = 10 1/4

Competition vs. Cooperation 500 Random networks Fully connected b = 1/4 Competition Cooperation

Bidirectionality 250 Random networks Fully connected b = 1/4 Opposition Reciprocity

Connectivity 250 Random networks N = 317, b = 1/4 DiluteFully connected 1%

Network Size 750 Random networks Fully connected b = 1/4 N = 317

What is the Smallest Chaotic Net? n dx 1 /dt = – bx 1 + tanh(x 4 – x 2 ) n dx 2 /dt = – bx 2 + tanh(x 1 + x 4 ) n dx 3 /dt = – bx 3 + tanh(x 1 + x 2 – x 4 ) n dx 4 /dt = – bx 4 + tanh(x 3 – x 2 ) Strange Attractor 2-torus

Circulant Networks dx i /dt = −bx i + Σ a j x i+j

Fully Connected Circulant Network N = 317

Diluted Circulant Network N = 317

Near-Neighbor Circulant Network N = 317

Summary of High- N Dynamics n Chaos is generic for sufficiently-connected networks n Sparse, circulant networks can also be chaotic (but the parameters must be carefully tuned) n Quasiperiodic route to chaos is usual n Symmetry-breaking, self-organization, pattern formation, and spatio-temporal chaos occur Maximum attractor dimension is of order N /2 n Attractor is sensitive to parameter perturbations, but dynamics are not

References n A paper on this topic is scheduled to appear soon in the journal Chaos n lectures/networks.ppt (this talk) lectures/networks.ppt n (my chaos textbook) n (contact me)