On homoclinic orbits to center manifolds of elliptic–hyperbolic equilibria in Hamiltonian systems

Abstract

We consider a Hamiltonian system which has an elliptic–hyperbolic equilibrium with a homoclinic loop. We identify the set of orbits which are homoclinic to the center manifold of the equilibrium via a Lyapunov–Schmidt reduction procedure. This leads to the study of a singularity which inherits a certain structure from the Hamiltonian nature of the system. Under non-degeneracy assumptions, we classify the possible Morse indices of this singularity, permitting a local description of the set of homoclinic orbits. We also consider the case of time-reversible Hamiltonian systems.