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 n Konstantinos Chlouverakis, Univ Athens (Greece)

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

4. Mathematics n The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. -- Eugene Wigner, 1960

5. 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)

6. 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

7. Example – The Weather

8. Bush-Kerry 2004 Race

9. Purple Map

10. Cellular Automaton X1X1 X2X2 X3X3 … …XNXN

11. Example – Game of Life n The Rules n For cells that are 'populated': u Each cell with one or no neighbors dies, as if by loneliness. u Each cell with four or more neighbors dies, as if by overpopulation. u Each cell with two or three neighbors survives. n For cell that are 'empty' (unpopulated): u Each empty cell with exactly three neighbors becomes populated.

12. Types of Models SpaceTimeStateModel Discrete CA Discrete ContinuousMaps DiscreteContinuousDiscrete ContinuousDiscrete Continuous ODEs ContinuousDiscreteContinuous Discrete Continuous PDE

13. Digital Computers n Computers are cellular automata n Space  RAM u 512 MBytes = 4 x 10 9 bits n Time  CPU speed u 2.4 GHz = 2.4 x 10 9 binary ops/second n State  Bus width u 32 bits = 4 x 10 9 distinct values n The program provides the rules whereby the CPU updates the RAM 256-bit core memory

14. Is Nature Discrete or Continuous? n No one knows n Some states are quantized u Electric charge, people, etc. n Quantum of space? u Planck length = 1.6 x 10 -35 meters n Quantum of time? u Planck time = 5.4 x 10 -44 seconds n Uncertainty principle u Δp Δx ~ 10 -35 kg-m/second u ΔE Δt ~ 10 -35 Joule-seconds

15. Characterization of Models n Intrinsic vs. extrinsic n Autonomous vs. nonautonomous n Variables vs. parameters n Stationary vs. nonstationary n Slow vs. fast time scales n Sensitive dependence vs. bifurcations n Stochastic vs. deterministic

16. Dimension n Spatial dimension (1, 2, 3) u Not a fundamental property u May provide guidance in choosing rules u Not relevant to all models (social networks, Web, financial markets, etc.) n Dynamical dimension (N) u Number of variables (sites, people, etc.) u Or number of required initial conditions u Reduces the problem to a single trajectory in N-dimensional space u May be infinite for continuous space 123… …N

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

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

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

20. What is the Simplest Complex NNet? n Complex = chaotic (positive LE) n Apparently 6 neurons u cf discrete-time systems with N = 2 u 6 x 5 = 30 connections u Can set 1 of 30 to zero u Cannot all be ±1 n With 7 neurons, all 42 can be ±1 n Apparently no symmetric cases

23. Real Electroencephlagrams

24. Strange Attractors

25. Sparse Circulant Network ( N=101)

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

28. Hyperlabyrinth Chaos ( N=101)

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

31. Lotka-Volterra Model ( N=101)

33. Delay Differential Equation

35. Partial Differential Equation

37. 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

38. Shameless Plug Chaos and Time-Series Analysis 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

39. References n http://sprott.physics.wisc.edu/ lectures/models.ppt (this talk) http://sprott.physics.wisc.edu/ lectures/models.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