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

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
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 12-01-0044113-01-12462 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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MSC: 37J, 37D, 70F
Accepted:01.12.2013
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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
2. S. V. Bolotin, “Degenerate billiards”, Proc. Steklov Inst. Math., 295 (2016), 45–62
3. Sergey V. Bolotin, “Degenerate Billiards in Celestial Mechanics”, Regul. Chaotic Dyn., 22:1 (2017), 27–53
4. S. V. Bolotin, “Jumps of energy near a separatrix in slow-fast Hamiltonian systems”, Russian Math. Surveys, 73:4 (2018), 725–727
5. Sergey V. Bolotin, “Jumps of Energy Near a Homoclinic Set of a Slowly Time Dependent Hamiltonian System”, Regul. Chaotic Dyn., 24:6 (2019), 682–703
6. Sergey V. Bolotin, “Local Adiabatic Invariants Near a Homoclinic Set of a Slow–Fast Hamiltonian System”, Proc. Steklov Inst. Math., 310 (2020), 12–24