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

TL;DR

This paper develops a method to analyze how small multivalued periodic perturbations affect the behavior of periodic solutions near a cycle in planar Hamiltonian systems, especially relevant for nonsmooth mechanical systems.

Contribution

It introduces a formula to estimate the deviation of perturbed solutions from a cycle in Hamiltonian systems with multivalued perturbations, extending classical analysis to nonsmooth contexts.

Findings

Provides a formula for the distance between unperturbed and perturbed solutions.

Applicable to nonsmooth mechanical systems with discontinuous periodic terms.

Assumes existence of a periodic solution for small perturbations.

Abstract

Aim of the paper is to provide a method to analyze the behavior of $T$-periodic solutions $x_\eps, \eps>0$, 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 $\eps\to0$. In several problems from nonsmooth mechanical systems this multivalued perturbation comes from the Filippov regularization of a nonlinear discontinuous $T$-periodic term. \noindent Through the paper, assuming the existence of a $T$-periodic solution $x_\eps$ for $\eps>0$ 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 $x_\eps([0,T])$ along a transversal direction to $x_0(t).$