## DYNAMICAL SYSTEMS Seminar

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

Abstract: Consider a Hamiltonian flow on R4 with a hyperbolic equilibrium $O$ and a transverse homoclinic orbit $\\\\Gamma$. In this talk, we discuss 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 introduce 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 show that near any orbit of this intersection, called super-homoclinic, there exist infinitely many multi-pulse homoclinic loops.