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Chaotic Dynamics on Large Networks

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

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Chaotic Dynamics on Large Networks

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  1. 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

  2. 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

  3. A Physicist’s Neuron N inputs tanh x x

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

  5. Examples of Networks Other examples: War, religion, epidemics, organizations, …

  6. Political System Information from others Political “state” a1 Voter a2 a3 aj = ±1/√N, 0 tanh x Democrat x Republican

  7. Types of Dynamics • Static • Periodic • Chaotic Equilibrium “Dead” Limit Cycle (or Torus) “Stuck in a rut” Strange Attractor Arguably the most “healthy” Especially if only weakly so

  8. Route to Chaos at Large N (=317) 400 Random networks Fully connected “Quasi-periodic route to chaos”

  9. Typical Signals for Typical Network

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

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

  12. Strange Attractors 10 1/4 N = b =

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

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

  15. Connectivity 250 Random networks N = 317, b = 1/4 Dilute Fully connected 1%

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

  17. 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 Attractor 2-torus

  18. Circulant Networks dxi /dt = −bxi + Σajxi+j

  19. Fully Connected Circulant Network N = 317

  20. Diluted Circulant Network N = 317

  21. Near-Neighbor Circulant Network N = 317

  22. 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

  23. 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) References

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