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

TL;DR

This paper analyzes the structure of homoclinic orbits to center manifolds in Hamiltonian systems with elliptic-hyperbolic equilibria, classifying their Morse indices and exploring reversible cases.

Contribution

It introduces a Lyapunov-Schmidt reduction approach to identify homoclinic orbits and classifies the singularities' Morse indices under non-degeneracy assumptions.

Findings

Classification of Morse indices of singularities

Local description of homoclinic orbit set

Extension to time-reversible 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 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.