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

### Oleg Makarenkov

Imperial College London, UK### Luisa Malaguti

Università di Modena e Reggio Emilia, Italy### Paolo Nistri

Universita di Siena, Italy

## 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. \noindent 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).

## Cite this article

Oleg Makarenkov, Luisa Malaguti, Paolo Nistri, On the Behavior of Periodic Solutions of Planar Autonomous Hamiltonian Systems with Multivalued Periodic Perturbations. Z. Anal. Anwend. 30 (2011), no. 2, pp. 129–144

DOI 10.4171/ZAA/1428