By K. Tomita (auth.), Professor Dr. Yoshiki Kuramoto (eds.)

ISBN-10: 3642695590

ISBN-13: 9783642695599

ISBN-10: 3642695612

ISBN-13: 9783642695612

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Additional info for Chaos and Statistical Methods: Proceedings of the Sixth Kyoto Summer Institute, Kyoto, Japan September 12–15, 1983

Sample text

The fractal dimension D of IF* tends toward 2 as planar chaos is approached, and the factor ~ tends to 1. 6. Second Path Beyond the Great Wave. A Scenario of Spiraling Towards Chaos (Fig. 4) Now select A to be within the atom I A-1/2 I < 1/2 but extremely close to A = 1. The IF* set is illustrated by Fig. 4. It is clear that as A-I, D - 2, hence ~ -1, and that chaos is approached without bifurcation. The facts are perhaps easier to visualize in terms of the parameter Po = 1/A and the variable u = 1/z.

Introduction Recently much attention has been paid to the relation between the integrabi I ity (non-integrabi I ity ) of dynamical systems and the distribution of singularities of solution in the complex t plane. Among others. CHANG et al. 1 ),2) found numerically in the Henon-Hei les system that the set of singularities have a self-simi lar structure and form natural boundaries of fractal dimension (0)1 ) roughly paral lei to the real t axis. e. (i) Why do the set of singularities have the self-simi lar structure?

1)Ashows that when o < OF the value of Ao converges exponentially while all other Am approach with liN corrections. When 0 > OF all eigenvalues approach 1 with liN corrections. Table 2. Convergence of OF when p = 1/4. 1) and N is the largest size lattice used in the fit. t the spectrum may actually be continuous between 0 and 1 with an isolated ~o > 1 w~en < 0*. The Pade analysis could easily miss this fact. It is interesting to note that liN decays have been observed in other dynamical systems which are hyperbolic except at isolated points [12].

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Chaos and Statistical Methods: Proceedings of the Sixth Kyoto Summer Institute, Kyoto, Japan September 12–15, 1983 by K. Tomita (auth.), Professor Dr. Yoshiki Kuramoto (eds.)


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