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Regul. Chaotic Dyn., 2000, Volume 5, Issue 2, Pages 139–156 (Mi rcd868)  

This article is cited in 1 scientific paper (total in 1 paper)

Infinite Number of Homoclinic Orbits to Hyperbolic Invariant Tori of Hamiltonian Systems

S. V. Bolotin

Department of Mathematics and Mechanics, Moscow State University, Vorob'ievy Gory, 119899, Moscow, Russia

Abstract: A time-periodic Hamiltonian system on a cotangent bundle of a compact manifold with Hamiltonian strictly convex and superlinear in the momentum is studied. A hyperbolic Diophantine nondegenerate invariant torus $N$ is said to be minimal if it is a Peierls set in the sense of the Aubry–Mather theory. We prove that $N$ has an infinite number of homoclinic orbits. For any family of homoclinic orbits the first and the last intersection point with the boundary of a tubular neighborhood $U$ of $N$ define sets in $U$. If there exists a compact family of minimal homoclinics defining contractible sets in $U$, we obtain an infinite number of multibump homoclinic, periodic and chaotic orbits. The proof is based on a combination of variational methods of Mather and a generalization of Shilnikov's lemma.

DOI: https://doi.org/10.1070/RD2000v005n02ABEH000137


Bibliographic databases:

MSC: 58F05, 58F08
Received: 01.03.2000
Language:

Citation: S. V. Bolotin, “Infinite Number of Homoclinic Orbits to Hyperbolic Invariant Tori of Hamiltonian Systems”, Regul. Chaotic Dyn., 5:2 (2000), 139–156

Citation in format AMSBIB
\Bibitem{Bol00}
\by S. V. Bolotin
\paper Infinite Number of Homoclinic Orbits to Hyperbolic Invariant Tori of Hamiltonian Systems
\jour Regul. Chaotic Dyn.
\yr 2000
\vol 5
\issue 2
\pages 139--156
\mathnet{http://mi.mathnet.ru/rcd868}
\crossref{https://doi.org/10.1070/RD2000v005n02ABEH000137}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1780706}
\zmath{https://zbmath.org/?q=an:1004.70018}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Alexey V. Ivanov, “Connecting Orbits near the Adiabatic Limit of Lagrangian Systems with Turning Points”, Regul. Chaotic Dyn., 22:5 (2017), 479–501  mathnet  crossref
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