Shilnikov lemma for a nondegenerate critical manifold of a hamiltonian system

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 tra jectories near M . Using this result, tra jectories 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 Poincar ́e’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 Poincarfor the unrestricted 3 body problem.