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Dynamics of High-Dimensional Systems

Dynamics of High-Dimensional Systems. J. C. Sprott Department of Physics University of Wisconsin - Madison Presented at the Santa Fe Institute On July 27, 2004. Collaborators. David Albers , SFI & U. Wisc - Physics Dee Dechert , U. Houston - Economics John Vano , U. Wisc - Math

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Dynamics of High-Dimensional Systems

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  1. Dynamics of High-Dimensional Systems J. C. Sprott Department of Physics University of Wisconsin - Madison Presented at the Santa Fe Institute On July 27, 2004

  2. Collaborators • David Albers, SFI & U. Wisc - Physics • Dee Dechert, U. Houston - Economics • John Vano, U. Wisc - Math • Joe Wildenberg, U. Wisc - Undergrad • Jeff Noel, U. Wisc - Undergrad • Mike Anderson, U. Wisc - Undergrad • Sean Cornelius,U. Wisc - Undergrad • Matt Sieth, U. Wisc - Undergrad

  3. Typical Experimental Data 5 x -5 500 0 Time

  4. 1 How common is chaos? Logistic Map xn+1 = Axn(1 −xn) Lyapunov Exponent -1 -2 A 4

  5. A 2-D Example (Hénon Map) 2 b xn+1 = 1 + axn2 + bxn-1 −2 a −4 1

  6. General 2-D Iterated Quadratic Map xn+1 = a1 + a2xn + a3xn2 + a4xnyn + a5yn + a6yn2 yn+1 = a7 + a8xn + a9xn2 + a10xnyn + a11yn + a12yn2

  7. General 2-D Quadratic Maps 100 % Bounded solutions 10% Chaotic solutions 1% 0.1% amax 0.1 1.0 10

  8. High-Dimensional Quadratic Maps and Flows Extend to higher-degree polynomials...

  9. Probability of Chaotic Solutions 100% Iterated maps 10% Continuous flows (ODEs) 1% 0.1% Dimension 1 10

  10. Correlation Dimension 5 Correlation Dimension 0.5 1 10 System Dimension

  11. Lyapunov Exponent 10 1 Lyapunov Exponent 0.1 0.01 1 10 System Dimension

  12. Neural Net Architecture tanh

  13. % Chaotic in Neural Networks D

  14. Attractor Dimension N = 32 DKY = 0.46 D D

  15. Routes to Chaos at Low D

  16. Routes to Chaos at High D

  17. Multispecies Lotka-Volterra Model • Let xi be population of the ith species (rabbits, trees, people, stocks, …) • dxi/ dt = rixi (1 − Σ aijxj ) • Parameters of the model: • Vector of growth rates ri • Matrix of interactions aij • Number of species N N j=1

  18. Parameters of the Model Growth rates Interaction matrix 1 a12a13a14a15a16 a21 1 a23a24a25a26 a31a32 1a34a35a36 a41a42a43 1a45a46 a51a52a53a54 1a56 a61a62a63a64a65 1 1 r2 r3 r4 r5 r6

  19. Choose riand aij randomly from an exponential distribution: 1 P(a) = e−a P(a) a = − LOG(RND) mean = 1 0 a 0 5

  20. Typical Time History 15 species xi Time

  21. Probability of Chaos • One case in 105 is chaotic for N = 4 with all species surviving • Probability of coexisting chaos decreases with increasing N • Evolution scheme: • Decrease selected aij terms to prevent extinction • Increase all aij terms to achieve chaos • Evolve solutions at “edge of chaos” (small positive Lyapunov exponent)

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

  23. Space Time

  24. Route to Chaos in Minimal LV Model

  25. Other Simple High-D Models

  26. Summary of High-D Dynamics • Chaos is the rule • Attractor dimension is ~ D/2 • Lyapunov exponent tends to be small (“edge of chaos”) • Quasiperiodic route is usual • Systems are insensitive to parameter perturbations

  27. http://sprott.physics.wisc.edu/ lectures/sfi2004.ppt (this talk) sprott@physics.wisc.edu (contact me) References

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