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04 December 2021
  » arxiv » 1308.4604

 Article overview

Shilnikov Lemma for a nondegenerate critical manifold of a Hamiltonian system
Sergey Bolotin ; Piero Negrini ;
Date 21 Aug 2013
AbstractWe prove an analog of Shilnikov Lemma for a normally hyperbolic symplectic critical manifold $Msubset H^{-1}(0)$ of a Hamiltonian system. Using this result, trajectories with small energy $H=mu>0$ shadowing chains of homoclinic orbits to $M$ are represented as extremals of a discrete variational problem, and their existence is proved. This paper is motivated by applications to the Poincar’e second species solutions of the 3 body problem with 2 masses small of order $mu$. As $mu o 0$, double collisions of small bodies correspond to a symplectic critical manifold of the regularized Hamiltonian system.
Source arXiv, 1308.4604
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