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Shilnikov lemma for a nondegenerate critical manifold of a Hamiltonian system

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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 λ-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 µ. As µ → 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.

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Dedicated to Alain Chenciner on occasion of his 70th birthday

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Bolotin, S., Negrini, P. Shilnikov lemma for a nondegenerate critical manifold of a Hamiltonian system. Regul. Chaot. Dyn. 18, 774–800 (2013). https://doi.org/10.1134/S1560354713060142

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

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