PERIODIC AND HOMOCLINIC ORBITS IN A TWO-PARAMETER UNFOLDING OF A HAMILTONIAN SYSTEM WITH A HOMOCLINIC ORBIT TO A SADDLE-CENTER

Abstract

For a two-degrees-of-freedom Hamiltonian system with a homoclinic orbit (loop) to a saddle-center we prove that the Poincaré map on a section to the loop within the Hamiltonian level containing a saddle-center is a twist map with discontinuity at the point of intersection with the loop. It explains the reason for existence of a countable set of periodic orbits near the loop. The types of these orbits are determined. We discover homoclinic doubling and tripling in a generic two-parameter Hamiltonian unfolding of such a Hamiltonian system. Besides this, we also study the nonautonomous linear Hamiltonian system obtained by means of the linearization of the original system at the loop. We derive the invariant formulation of the genericity condition that is needed for studying the original system.