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Simple Models of Complex Chaotic Systems

Simple Models of Complex Chaotic Systems. J. C. Sprott Department of Physics University of Wisconsin - Madison Presented at the AAPT Topical Conference on Computational Physics in Upper Level Courses At Davidson College (NC) On July 28, 2007. Background.

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Simple Models of Complex Chaotic Systems

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  1. Simple Models of Complex Chaotic Systems J. C. Sprott Department of Physics University of Wisconsin - Madison Presented at the AAPT Topical Conference on Computational Physics in Upper Level Courses At Davidson College (NC) On July 28, 2007

  2. Background • Grew out of an multi-disciplinary chaos course that I taught 3 times • Demands computation • Strongly motivates students • Used now for physics undergraduate research projects (~20 over the past 10 years)

  3. Minimal Chaotic Systems • 1-D map (quadratic map) • Dissipative map (Hénon) • Autonomous ODE (jerk equation) • Driven ODE (Ueda oscillator) • Delay differential equation (DDE) • Partial diff eqn (Kuramoto-Sivashinsky)

  4. 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 and adapt

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

  6. A General Model (artificial neural network) N neurons “Universal approximator,” N  ∞

  7. Route to Chaos at Large N (=101) “Quasi-periodic route to chaos”

  8. Strange Attractors

  9. Sparse Circulant Network (N=101)

  10. Labyrinth Chaos dx1/dt = sin x2 dx2/dt = sin x3 dx3/dt = sin x1 x1 x3 x2

  11. Hyperlabyrinth Chaos (N=101)

  12. Minimal High-D Chaotic L-V Model dxi /dt = xi(1 – xi –2– xi – xi+1)

  13. Lotka-Volterra Model (N=101)

  14. Delay Differential Equation

  15. Partial Differential Equation

  16. Summary of High-N Dynamics • Chaos is common for highly-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

  17. Shameless Plug Chaos and Time-Series Analysis J. C. SprottOxford University Press (2003) ISBN 0-19-850839-5 An introductory text for advanced undergraduate and beginning graduate students in all fields of science and engineering

  18. http://sprott.physics.wisc.edu/ lectures/models.ppt (this talk) http://sprott.physics.wisc.edu/chaostsa/ (my chaos textbook) sprott@physics.wisc.edu (contact me) References

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