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					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
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

 N                             N
                   xout  tanh  a j x j
inputs                         j1


          tanh x


                               x
     A General Model
(artificial neural network)

                      1
                                     N neurons
                      3
             2                    4

                              N
          xi  bi xi  tanh  aij x j
          
                              j 1
                              j i

        “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    Traders     Trans-       Wealth         Money
 Market                  actions


 Political   Voters    Information     Party       The Press
 System                              affiliation



Other examples: War, religion, epidemics, organizations, …
               Political System
Information
               a1               Political “state”
from others                                       N
                    Voter     x  bx  tanh  a j x j
                              
          a2                                      j 1
               a3             aj = ±1/√N, 0

                     tanh x             Democrat



                                              x
        Republican
Types of Dynamics

1.   Static      Equilibrium
     “Dead”
2.   Periodic Limit Cycle (or Torus)
     “Stuck in a rut”
3.   Chaotic Strange Attractor
     Arguably the most “healthy”
     Especially if only weakly so
Route to Chaos at Large N (=317)
                          317
   dxi / dt  bxi  tanh  aij x j
                          j1
                                        400 Random networks
                                           Fully connected




                                “Quasi-periodic route to chaos”
Typical Signals for Typical Network
  Average Signal from all Neurons
All +1                 N = 317
                       b = 1/4




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




100% Republican
Strange Attractors
       N = 10
       b = 1/4
Competition vs. Cooperation
                       317
dxi / dt  bxi  tanh  aij x j
                       j1
                                   500 Random networks
                                      Fully connected
                                          b = 1/4



  Competition



                                           Cooperation
              Bidirectionality
                       317
dxi / dt  bxi  tanh  aij x j
                       j1
                                   250 Random networks
                                      Fully connected
                                          b = 1/4




                                            Reciprocity


  Opposition
                 Connectivity
                        317
dxi / dt  bxi  tanh  aij x j
                        j1
                                   250 Random networks
                                     N = 317, b = 1/4


Dilute                                     Fully connected


                   1%
                Network Size
                        N
dxi / dt  bxi  tanh  aij x j
                       j1
                                   750 Random networks
                                      Fully 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)

                         Strange                     2-torus
                         Attractor
  Circulant Networks
dxi /dt = −bxi + Σ ajxi+j
Fully Connected Circulant Network
                          N 1
   dxi / dt  bxi  tanh  a j xi j
                          j1

   N = 317
Diluted Circulant Network
dxi / dt  bxi  tanh(xi42  xi126  xi254 )

N = 317
Near-Neighbor Circulant Network
   dxi / dt  bxi  tanh(xi1  xi2  xi3  xi4  xi5  xi6)

   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)

				
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