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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 30, Issue 2, 2011, pp. 129–144**

**DOI: 10.4171/ZAA/1428**

Published online: 2011-04-03

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

Oleg Makarenkov^{[1]}, Luisa Malaguti

^{[2]}and Paolo Nistri

^{[3]}(1) Imperial College London, UK

(2) Università di Modena e Reggio Emilia, Italy

(3) Universita di Siena, Italy

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).

*Keywords: *Planar Hamiltonian systems, characteristic multipliers, multivalued periodic perturbations, periodic solutions, approximation formula

Makarenkov Oleg, Malaguti Luisa, Nistri Paolo: On the Behavior of Periodic Solutions of Planar Autonomous Hamiltonian Systems with Multivalued Periodic Perturbations. *Z. Anal. Anwend.* 30 (2011), 129-144. doi: 10.4171/ZAA/1428