The Melnikov Method and Subharmonic Orbits in a Piecewise-Smooth System

We consider a two-dimensional piecewise-smooth system defined in two domains separated by a switching manifold Σ. 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 Σ. Finally, we assume that the region enclosed by these heteroclinic connections is fully covered by periodic orbits surrounding the origin, whose periods monotonically increase as they approach the heteroclinic connection. For a nonautonomous (T-periodic) Hamiltonian perturbation of amplitude ε, we rigorously prove, for every n and m relatively prime and ε>0 small enough, that there exists an $nT$-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 Σ, then all these orbits exist if the relative size of ε>0 with respect to the magnitude of this jump is large enough. In addition, we obtain similar conditions for the splitting of the heteroclinic connections.