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Invariant manifolds for near identity differentiable maps and splitting of separatrices

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 consider families of differentiable diffeomorphisms with hyperbolic points, close to the identity, which tend to it when the parameter goes to zero.

We study the asymptotic behaviour of the invariant manifolds. Then we consider the case when there are homo-heteroclinic points and we find that the maximum separation between the invariant manifolds is of the order of some power of the parameter which is related to the degree of differentiability.

Finally the analogous case for flows is considered.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 2 , June 1990 , pp. 319 - 346
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[1]Abraham, R., Marsden, J. E. & Ratiu, T.. Manifolds, Tensor Analysis, and Applications. Addison-Wesley: Massachusetts, 1983.Google Scholar
[2]Arnold, V. I.. Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surveys 18 (1963), 9–36.CrossRefGoogle Scholar
[3]Arnold, V. I. & Avez, A.. Problèmes Ergodiques de la Mécanique Classique. Gauthier Villars: Paris, 1967.Google Scholar
[4]Fontich, E. & Simó, C.. The splitting of separatrices for analytic diffeomorphisms. Preprint (1986) and this Volume.Google Scholar
[5]Guckenheimer, J. & Holmes, P.. Nonlinear Oscillations, Dynamical systems, and Bifurcations of Vector Fields. Springer: New York, 1983.Google Scholar
[6]Hartman, P.. Ordinary Differential Equations. 2nd Ed., Birkhäuser: Boston, 1982.Google Scholar
[7]Hirsch, M. & Pugh, C.. Stable manifolds and hyperbolic sets. Proc. Symp. in Pure Math 14, Amer. Math. Soc. (1970) 133–164.Google Scholar
[8]Lazutkin, V. F.. Splitting of separatrices for the Chirikov's standard map. Preprint VINITI 6372/84 (1984).Google Scholar
[9]Llibre, J. & Simó, C.. On the Hénon-Heiles Potential. Adas III CEDYA, Santiago de Compostela (1980), 183–206.Google Scholar
[10]Llibre, J. & Simó, C.. Oscillatory solutions in the planar restricted three body problem. Math. Ann. 248 (1980), 153–184.CrossRefGoogle Scholar
[11]McGehee, R. & Meyer, K.. Homoclinic points of area preserving diffeomorphisms. Amer. J. Math. 96 (1974), 409–421.CrossRefGoogle Scholar
[12]Sanders, J.. Melnikov's method and averaging. Cel. Mech. 28 (1982), 171–181.CrossRefGoogle Scholar
[13]Zehnder, E.. Homoclinic points near elliptic fixed points. Comm. PureAppl. Math. 26 (1973), 131–182.Google Scholar