On the behavior of periodic solutions of planar autonomous Hamiltonian systems with multivalued periodic perturbations

Aim of the paper is to provide a method to analyze the behavior of T-periodic solutions x(epsilon), epsilon > 0, of a perturbed planar Hamiltonian system near a cycle x(0), of smallest period T, of the unperturbed system. The perturbation is represented by a T-periodic multivalued map which vanishes as epsilon -> 0. In several problems from nonsmooth mechanical systems this multivalued perturbation comes from the Filippov regularization of a nonlinear discontinuous T-periodic term. Through the paper, assuming the existence of a T-periodic solution x(epsilon) for epsilon > 0 small, under the condition that x(0) is a nondegenerate cycle of the linearized unperturbed Hamiltonian system we provide a formula for the distance between any point x(0)(t) and the trajectories x(epsilon)([0, T)) along a transversal direction to x(0)(t).