Invariant manifolds of homoclinic orbits: super-homoclinics and multi-pulse homoclinic loops

Abstract

Consider a Hamiltonian flow on \(\mathbb{R}^{4}\) with a hyperbolic equilibrium \(O\) and a transverse homoclinic orbit \(\Gamma\). In this paper, we study the dynamics near \(\Gamma\) in its energy level when it leaves and enters \(O\) along strong unstable and strong stable directions, respectively. In particular, we provide necessary and sufficient conditions for the existence of the local stable and unstable invariant manifolds of \(\Gamma\). We then consider the case in which both of these manifolds exist. We globalize them and assume they intersect transversely. We prove that near any orbit of this intersection, called super-homoclinic, there exist infinitely many multi-pulse homoclinic loops.