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 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 the parameter tends to zero. In several problems from nonsmooth mechanical systems this multivalued perturbation comes from the Filippovregularization of a nonlinear discontinuous T-periodic term. Through the paper, assuming the existence of a T-periodic solution of the parameterized problems, under the conditionthat 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 of the perturbed period solutions along a transversal direction to x_0(t).