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Chaos Mappings

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Chaos Mappings

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    1. Chaos Mappings

    2. Flows vs. Maps System of ordinary first-order differential equations: where t = independent variable xi º x1, x2, . . . , xn. example of a flow

    3. Flows Flow – gives rise to continuous evolution of “field lines” in the n-dimensional space (or “phase space”) If the volume in space remains constant with time, the flow is conservative. If the volume in space decreases with time, the flow is dissipative. A dampening effect in dynamics like friction

    4. Flows The first system of equations can be written in column vector form:

    5. Lorenz Equations Chaotic effects arises when At least one of the functions fi contains a nonlinear term (e.g., x12, x12x2, x1x23) The dimension of the system of equations is 3 or greater s, r, and b are all constants Lorenz attractor – when s = 10, b = 8/3, and r = 28

    6. Dynamic Systems as Maps XN+1 = G(XN), N = 1, 2, . . . can be defined by the column vectors where N labels the Nth iteration of the map

    7. Dynamic Systems as Maps A map can be generated from a flow by taking: X(t), X(t + t), X(t + 2t), . . . X0, X1, X2, . . . Condition for chaos in mappings Must contain at least one nonlinear term

    8. Flows vs. Maps

    9. Simple Maps The logistic map The Hénon attractor Chaos esthétique The Standard Map

    10. The Logistic Map Xn+1 = axn(1-xn) At iteration 1000 1.0 – below this value the population cannot survive 2.0 – oscillatory approach to the asymptotic value 3.0 – “period” of the population doubles 3.45 – “something else happens”

    11. The Logistic Map Adding iteration 1001 At around 3.57 – chaos emerges Chaos does not necessarily imply disorder Chaos is the “randomness” in predicting the next iteration

    12. The Logistic Map Adding iteration 1003 Period quadruples at 3.449499

    13. The Logistic Map 256 iterations after i1000 3.544090 – period of 8 3.564407 – period of 16 3.568759 – period of 32 3.569692 – period of 64 3.569946 – period doubling ends

    14. The Logistic Map One of the branches is a small replication of the entire function “Self similarity” across scales

    15. The Logistic Map Y range = 0.489 – 0.52 X range = 3.625 – 3.638 Box: 3.6339 – 3.6342

    16. The Logistic Map Y range = 0.491 – 0.501 Box = 3.634042 – 3.634052

    17. The Logistic Map Y range = 0.499621 – 0.50015 X range = 3.63404761 – 3.63404998 Magnification nearly 1 million times that of the first chaos mapping

    18. The Logistic Map ~ 3.569946 – period doubling region ends and chaos begins 3.828427 – small period tripling window opens up ~ 3.855 – period tripling cascade ends and chaos resumes ~ 4.0 chaos reigns!!!

    19. The Logistic Map Both periodicity and chaos in this picture 3.828427 – small period tripling window opens up ~ 3.855 – period tripling cascade ends and chaos resumes

    20. The Hénon Attractor 2-D map given by the equations: xn+1 = yn + 1 – axn2 yn+1 = Bxn General form of the attractor does not depend on initial x and y values

    21. The Hénon Attractor

    22. The Hénon Attractor Data generated through C++ Rendered with POVray 24bit undersampled 640x480 image 10 – 12 hours to render http://www.ph.utexas.edu/~morrow/Henon/henon.html

    23. The Chaos Esthétique 2-D mapping for modeling the dynamics of a particle accelerator xn+1 = yn + f(xn) yn+1 = -bxn + f(xn+1) where a and b are constants and f(x) = ax + [2(1 – a)x2 / (1 + x2)]

    24. Conservative Mapping

    25. Dissipative Mapping

    26. The Standard Map 2-D map to model accelerator dynamics qn+1 = qn + pn+1 pn+1 = pn + (k/2p) sin(2pqn) for small values of k there is no chaos for values of k above ~ 4 chaos reigns the onset of widespread chaotic behavior occurs ~ 0.9716

    27. The Standard Map closed loops – stable regions with fixed or periodic points at the centers hazy regions – unstable and chaotic

    28. The Standard Map

    29. References D. Gulick, “Encounters with Chaos” (McGraw Hill, Inc., New York, 1992), pp. 127-186, 195-220, 240-285 P. Berge, Y. Pomeau, and C. Vidal, “Order Within Chaos; Towards a Deterministic Approach to Turbulence” (John Wiley & Sons, New York, 1984), pp. 111-144, 301-324. R. Devaney, “A First Course in Chaotic Dynamical Systems” (Addison-Wesley Publishing Company, Inc., New York, 1992), pp. 154-163. H. Lauwerier, “Fractals: Endlessly Repeating Geometrical Figures” (Princeton University Press, Princeton, N.J., 1991), p. 136. M. Tabor, “Chaos and Integrability in Nonlinear Dynamics, An Introduction” (John Wiley & Sons, New York, 1989), pp. 134-167. K. T. R. Davies and M. Baranger, to be published.

    30. Web References Exploring the Logistic Map – M. Casco Associates Strange attractors – Henon, etc. Standard Map - Cirikov-Taylor map Heun attractor program in BASIC

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