Dynamical Features in a Slow-fast Piecewise Linear Hamiltonian System

Abstract

We demonstrate that a piecewise linear slow-fast Hamiltonian system with an equi- librium of the saddle-center type can have a sequence of small parameter values for which a one-round homoclinic orbit to this equilibrium exists. This contrasts with the well-known find- ings by Amick and McLeod and others that solutions of such type do not exist in analytic Hamiltonian systems, and that the separatrices are split by the exponentially small quantity. We also discuss existence of homoclinic trajectories to small periodic orbits of the Lyapunov family as well as symmetric periodic orbits near the homoclinic connection. Our further re- sult, illustrated by simulations, concerns the complicated structure of orbits related to passage through a non-smooth bifurcation of a periodic orbit.