Type I periodic motions for nonlinear impact oscillators

In this paper a general class of nonlinear impact oscillators is considered for subharmonic bifurcation. This system can be used to model an inverted pendulum impacting on rigid walls under external periodic excitation and its unperturbed system possesses a pair of homoclinic cycles via the identification given by the impact law and three separate families of periodic orbits inside and outside the homoclinic cycles. By discussing the subharmonic orbits inside the homoclinic cycles, the subharmonic Melnikov method established for smooth dynamical systems is extended to be applicable to the non-smooth system.