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 $\Sigma$. 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 $\Sigma$. 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 $\varepsilon$, we rigorously prove, for every $n$ and $m$ relatively prime and $\varepsilon>0$ 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 $\Sigma$, then all these orbits exist if the relative size of $\varepsilon>0$ with respect to the magnitude of this jump is large enough. In addition, we obtain similar conditions for the splitting of the heteroclinic connections.