On the Behavior of Periodic Solutions of Planar Autonomous Hamiltonian Systems with Multivalued Periodic Perturbations

Abstract

Aim of the paper is to provide a method to analyze the behavior of T -periodic solutions x_\varepsilon , \varepsilon>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 \varepsilon\to 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_\varepsilon for \varepsilon>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_\varepsilon([0,T]) along a transversal direction to x_0(t) .