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Regul. Chaotic Dyn., 2013, Volume 18, Issue 6, Pages 774–800 (Mi rcd169)  

This article is cited in 4 scientific papers (total in 4 papers)

Shilnikov Lemma for a Nondegenerate Critical Manifold of a Hamiltonian System

Sergey Bolotinab, Piero Negrinic

a V. A. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
b University of Wisconsin–Madison, 480 Lincoln Dr., Madison, WI 53706-1325, USA
c Dipartimento di Matematica, Sapienza, Università di Roma, Piazzale Aldo Moro 5, 00185 Rome, Italy

Abstract: Let $M$ be a normally hyperbolic symplectic critical manifold of a Hamiltonian system. Suppose $M$ consists of equilibria with real eigenvalues. We prove an analog of the Shilnikov lemma (strong version of the $\lambda$-lemma) describing the behavior of trajectories near $M$. Using this result, trajectories shadowing chains of homoclinic orbits to $M$ are represented as extremals of a discrete variational problem. Then the existence of shadowing periodic orbits is proved. This paper is motivated by applications to the Poincaré’s second species solutions of the $3$ body problem with $2$ masses small of order $\mu$. As $\mu \to 0$, double collisions of small bodies correspond to a symplectic critical manifold $M$ of the regularized Hamiltonian system. Thus our results imply the existence of Poincaré’s second species (nearly collision) periodic solutions for the unrestricted $3$ body problem.

Keywords: Hamiltonian system, symplectic map, generating function, heteroclinic orbit

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-00441
13-01-12462
The work of S. Bolotin was supported by the Programme “Dynamical Systems and Control Theory” of RAS and RFBR grants ¹ 12-01-00441 and ¹ 13-01-12462.


DOI: https://doi.org/10.1134/S1560354713060142

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Bibliographic databases:

Document Type: Article
MSC: 37J, 37D, 70F
Received: 31.07.2013
Accepted:01.12.2013
Language: English

Citation: Sergey Bolotin, Piero Negrini, “Shilnikov Lemma for a Nondegenerate Critical Manifold of a Hamiltonian System”, Regul. Chaotic Dyn., 18:6 (2013), 774–800

Citation in format AMSBIB
\Bibitem{BolNeg13}
\by Sergey Bolotin, Piero Negrini
\paper Shilnikov Lemma for a Nondegenerate Critical Manifold of a Hamiltonian System
\jour Regul. Chaotic Dyn.
\yr 2013
\vol 18
\issue 6
\pages 774--800
\mathnet{http://mi.mathnet.ru/rcd169}
\crossref{https://doi.org/10.1134/S1560354713060142}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3146592}
\zmath{https://zbmath.org/?q=an:06292773}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000329108900014}


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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. S. V. Bolotin, D. V. Treschev, “The anti-integrable limit”, Russian Math. Surveys, 70:6 (2015), 975–1030  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. S. V. Bolotin, “Degenerate billiards”, Proc. Steklov Inst. Math., 295 (2016), 45–62  mathnet  crossref  crossref  mathscinet  isi  elib
    3. Sergey V. Bolotin, “Degenerate Billiards in Celestial Mechanics”, Regul. Chaotic Dyn., 22:1 (2017), 27–53  mathnet  crossref  mathscinet
    4. S. V. Bolotin, “Jumps of energy near a separatrix in slow-fast Hamiltonian systems”, Russian Math. Surveys, 73:4 (2018), 725–727  mathnet  crossref  crossref  adsnasa  isi  elib
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