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The splitting of separatrices for analytic diffeomorphisms

Published online by Cambridge University Press:  19 September 2008

E. Fontich  and
C. Simó
E. Fontich
Affiliation:
Dept. de Matemàtica Aplicada, E.T.S.E.I.B., University Politècnica de Catalunya, Barcelona, Spain
C. Simó
Affiliation:
Dept. de Matemàtica Aplicada i Anàlisi, University de Barcelona, Barcelona, Spain
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Abstract

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We study families of diffeomorphisms close to the identity, which tend to it when the parameter goes to zero, and having homoclinic points. We consider the analytical case and we find that the maximum separation between the invariant manifolds, in a given region, is exponentially small with respect to the parameter. The exponent is related to the complex singularities of a flow which is taken as an unperturbed problem. Finally several examples are given.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 2 , June 1990 , pp. 295 - 318
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[1]Braun, M.. On the applicability of the third integral of motion. J. Diff. Eq. 13 (1973), 300–318.CrossRefGoogle Scholar
[2]Churchill, R., Pecelli, G. & Rod, D.. A survey of the Hénon-Heiles Hamiltonian with applications to related examples. In Lecture Notes in Physics 93 (1979) 76–136.Google Scholar
[3]Cornfeld, I. P. & Sinai, Ya. G.. Splitting of separatrices for the standard mapping. Unpublished paper (1982).Google Scholar
[4]Fontich, E.. On integrability of quadratic area preserving mappings in the plane. In Lecture Notes in Physics 179 (1983), 270–271.Google Scholar
[5]Fontich, E. & Simó, C.. Invariant manifolds for near identity differentiable maps and splitting of separatrices. This issue.Google Scholar
[6]Guckenheimer, J. & Holmes, P.. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer: Berlin-Heidelberg-New York-Tokyo, 1983.CrossRefGoogle Scholar
[7]Gustavson, F. G.. On constructing formal integrals of a Hamiltonian system near an equilibrium point. Astron. J. 71 (1966), 670–686.CrossRefGoogle Scholar
[8]Hénon, M. & Heiles, C.. The applicability of the third integral of motion, some numerical experiments. Astron. J. 69 (1964), 73–79.CrossRefGoogle Scholar
[9]Holmes, P., Marsden, J. & Scheurle, J.. Exponentially small splitting of separatrices. (1987). Preprint.Google Scholar
[10]Lazutkin, V. F.. Splitting of separatrices for the Chirikov's standard map. Preprint VINITI 6372/84 (1984).Google Scholar
[11]Lazutkin, V. F.. Private communication (1985).Google Scholar
[12]Llibre, J. & Simó, C.. On the Hénon-Heiles potential. Adas del III CEDYA. Santiago de Compostela (1980), 183–206.Google Scholar
[13]McGehee, R. & Meyer, K.. Homoclinic points of area preserving diffeomorphisms. Amer. J. Math. 96 (1974), 409–421.CrossRefGoogle Scholar
[14]Melnikov, V. K.. On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc. 12 (1963), 3–56.Google Scholar
[15]Moser, J.. The analytical invariants of an area preserving mapping near a hyperbolic fixed point. Comm. Pure App. Math. 9 (1956), 673–692.CrossRefGoogle Scholar
[16]Moser, J.. Lectures on Hamiltonian Systems. Memoirs Amer. Math. Soc. 81 (1968), 1–60.Google Scholar
[17]Sanders, J.. Melnikov's method and averaging. Cel. Mech. 28 (1982), 171–181.CrossRefGoogle Scholar
[18]Siegel, C. & Moser, J.. Lectures on Celestial Mechanics. Springer: Berlin-Heidelberg-NewYork, 1971.CrossRefGoogle Scholar