The Melnikov method and subharmonic orbits in a piecewise smooth system

Abstract

In this work we consider a two-dimensional piecewise smooth system, defined in two domains separated by the switching manifold x = 0. We assume that there exists a piecewise-defined continuous Hamiltonian that is a first integral of the system. We also suppose that the system possesses an invisible fold-fold at the origin and two heteroclinic orbits connecting two hyperbolic critical points on either side of x = 0. Finally, we assume that the region closed by these heteroclinic connections is fully covered by periodic orbits surrounding the origin, whose periods monotonically increase as they approach the heteroclinic connection. When considering a non-autonomous (T-periodic) Hamiltonian perturbation of amplitude ", using an impact map, we rigorously prove that, for every n and m relatively prime and " > 0 small enough, there exists a nT-periodic orbit impacting 2m times with the switching manifold at every period if a modified subharmonic Melnikov function possesses a simple zero. We also prove that, if the orbits are discontinuous when they cross x = 0, then all these orbits exist if the relative size of " > 0 with respect to the magnitude of this jump is large enough. We also obtain similar conditions for the splitting of the heteroclinic connections.