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 $x_{\epsilon }$, $\epsilon >0$, 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 $\epsilon \to 0$. 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 $x_{\epsilon }$ 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 $x_0(t)$.