Horseshoes in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers

Abstract:

We consider a class of two-degree-of-freedom Hamiltonian systems which have saddle-centers with homoclinic orbits and do not take the form of small perturbations of integrable systems. Using a Melnikov-type global perturbation technique, we present a condition under which orbits transversely homoclinic to periodic orbits exist and horseshoes are created on surfaces of energy level near the saddle-centers. We apply the theory to systems with potentials and give concrete examples. In particular, we can detect the presence of horseshoes in a class of systems to which previous results did not apply.