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. Collaborators n David Albers, Univ California - Davis David Albers n Konstantinos Chlouverakis, Univ Athens (Greece) Konstantinos Chlouverakis

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

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

5. 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 and adapt

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

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

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

9. Strange Attractors

10. Sparse Circulant Network ( N=101)

12. Labyrinth Chaos x1x1 x3x3 x2x2 dx 1 /dt = sin x 2 dx 2 /dt = sin x 3 dx 3 /dt = sin x 1

13. Hyperlabyrinth Chaos ( N=101)

15. Minimal High-D Chaotic L-V Model dx i /dt = x i (1 – x i – 2 – x i – x i + 1 )

16. Lotka-Volterra Model ( N=101)

18. Delay Differential Equation

20. Partial Differential Equation

22. Summary of High- N Dynamics n Chaos is common for highly-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

23. Shameless Plug Chaos and Time-Series Analysis J. C. Sprott J. C. Sprott Oxford 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

24. References n http://sprott.physics.wisc.edu/ lectures/davidson.ppt (this talk) http://sprott.physics.wisc.edu/ lectures/davidson.ppt n http://sprott.physics.wisc.edu/chao stsa/ (my chaos textbook) http://sprott.physics.wisc.edu/chao stsa/ n sprott@physics.wisc.edu (contact me) sprott@physics.wisc.edu