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

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
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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} 

• http://mi.mathnet.ru/eng/rcd868
• http://mi.mathnet.ru/eng/rcd/v5/i2/p139

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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