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

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Aim of the paper is to provide a method to analyze the behavior of T-periodic solutions x, >0, of a perturbed planar Hamiltonian system near a cycle x₀, of smallest period T, of the unperturbed system. The perturbation is represented by a T-periodic multivalued map which vanishes as →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 for >0 small, under the condition that x₀ is a nondegenerate cycle of the linearized unperturbed Hamiltonian system we provide a formula for the distance between any point x₀(t) and the trajectories x([0,T]) along a transversal direction to x₀(t).