Download e-book for kindle: Chaos: A Program Collection for the PC by Professor Dr. H. J. Korsch, Professor Dr. H.-J. Jodl (auth.)

By Professor Dr. H. J. Korsch, Professor Dr. H.-J. Jodl (auth.)

ISBN-10: 366202991X

ISBN-13: 9783662029916

ISBN-10: 3662029936

ISBN-13: 9783662029930

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This is, however, not the case. The transition from an integrable to a nonintegrable system is most clearly analyzed in one of the most fundamental results in the theory of Hamiltonian (conservative) systems: the celebrated theorem of Kolmogoroff, Arnold, and Moser, which describes the infiuence of perturbations on an integrable system. Here, we formulate this so-called 'KAM theorem' for two degrees of freedom. 21 ) H(p, q) = Ho(p, q) + cH1(p, q) (the mathematical proof requires certain differentiability properties of H 1 ).

Jorna, editor, Topics in Nonlinear Dynamies, page 16. Am. Inst. Phys. Conf. Proc. Vol, 46 1978 (reprinted in R. S. MacKay and J. D. Meiss, Hamiltonian Dynamical Systems, Adam Hilger, Bristol 1987. K. R. Meyer, Generic bifurcations 01 periodic points, Trans. AMS 149 (1970) 95 J. M. Greene, R. S. MacKay, F. Vivaldi, and M. J. Feigenbaum, Universal behaviour in families 01 area-preserving maps, Physica D 3 (1981) 468 (reprinted in: R. S. MacKay and J. D. Meiss, Hamiltonian Dynamical Systems, Adam Hilger, Bristol 1987.

83) for x* = X n , n = 0, ... , k - l. 83) that the slope of fk is identical for all k members x n , n = 0, ... , k - 1 of the k-periodic orbit. Graphically, this means that the function fk(X) becomes simultaneously tangential to the bisector y(x) = x for the whole chain of fixed points when this orbit loses stability. Let us look at the loss of stability of a fixed point x* = f (x* , r) in more detail. Increasing the parameter T from a stable region ( If' (x* , r) I < 1) to an unstable region (If'(x*, T)I > 1) crossing a critical value Tl with If'(x*, Tl)1 = 1, the fixed point x* = X*(T) becomes unstable at T = Tl.

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Chaos: A Program Collection for the PC by Professor Dr. H. J. Korsch, Professor Dr. H.-J. Jodl (auth.)


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