Topologically transversal reversible homoclinic sets

Fečkan, Michal

doi:10.1090/S0002-9939-02-06473-0

Topologically transversal reversible homoclinic sets
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by Michal Fečkan PDF

Proc. Amer. Math. Soc. 130 (2002), 3369-3377 Request permission

Abstract:

An $R$-reversible diffeomorphism on $\mathbb {R}^{2N}$ is studied possessing a hyperbolic fixed point. If the stable manifold of the hyperbolic fixed point and the fixed point set $\operatorname {Fix} R$ of $R$ have a nontrivial local topological crossing, then an infinite number of $R$-symmetric periodic orbits of the diffeomorphism is shown. A perturbed problem is also studied by showing the relationship between a corresponding Melnikov function and the nontriviality of a local topological crossing of the set $\operatorname {Fix} R$ and the stable manifold for the perturbed diffeomorphism.

References

Battelli, F., Fečkan, M., Chaos arising near a topologically transversal homoclinic set, preprint.
Flaviano Battelli and Michal Fe kan, Subharmonic solutions in singular systems, J. Differential Equations 132 (1996), no. 1, 21–45. MR 1418498, DOI 10.1006/jdeq.1996.0169
Robert L. Devaney, Blue sky catastrophes in reversible and Hamiltonian systems, Indiana Univ. Math. J. 26 (1977), no. 2, 247–263. MR 431274, DOI 10.1512/iumj.1977.26.26018
Robert L. Devaney, Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc. 218 (1976), 89–113. MR 402815, DOI 10.1090/S0002-9947-1976-0402815-3
Robert L. Devaney, Homoclinic bifurcations and the area-conserving Hénon mapping, J. Differential Equations 51 (1984), no. 2, 254–266. MR 731153, DOI 10.1016/0022-0396(84)90110-4
Irene Fonseca and Wilfrid Gangbo, Degree theory in analysis and applications, Oxford Lecture Series in Mathematics and its Applications, vol. 2, The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications. MR 1373430
Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362
Kenneth J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, Dynamics reported, Vol. 1, Dynam. Report. Ser. Dynam. Systems Appl., vol. 1, Wiley, Chichester, 1988, pp. 265–306. MR 945967
Vanderbauwhede, A., Heteroclinic cycles and periodic orbits in reversible systems, in J. Wiener & J.K. Hale (Ed.) “Ordinary and Delay Differential Equations”, Pitman Research Notes in Mathematics Series, No. 272 Pitman 1992, pp. 250-253.
André Vanderbauwhede and Bernold Fiedler, Homoclinic period blow-up in reversible and conservative systems, Z. Angew. Math. Phys. 43 (1992), no. 2, 292–318. MR 1162729, DOI 10.1007/BF00946632