Albert Granados (Universität Stuttgart): Melnikov method for subharmonic orbits and heteroclinic connections in a non-smooth system

In this work we consider a two-dimensional piecewise smooth system,
defined in two sets separated by the switching curve 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 fold-fold at the origin and two heteroclinic
orbits connecting two critical saddle points located at each 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 eps, using an impact Poincaré map, we
rigorously prove that, for every n and m relatively prime and
eps>0 small enough, there exists a nT-periodic orbit impacting
2m times with the switching curve at every period if a modified
subharmonic Melnikov function possesses a simple zero. In addition,
we also prove that, if the orbits are forced to undergo a
discontinuity when they cross x=0, which simulates a loss of energy,
then all these orbits persist if the relative size of eps>0 with
respect to the magnitude of this jump is large enough.

We also obtain similar conditions for the splitting of the
separatrices.