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 TT-periodic solutions xϵ,ϵ>0x_\epsilon, \, \epsilon>0, of a perturbed planar Hamiltonian system near a cycle~x0x_0, of smallest period TT, of the unperturbed system. The perturbation is represented by a TT-periodic multivalued map which vanishes as ϵ0\epsilon\to 0. In several problems from nonsmooth mechanical systems this multivalued perturbation comes from the Filippov regularization of a nonlinear discontinuous TT-periodic term. \noindent Through the paper, assuming the existence of a TT-periodic solution xϵx_\epsilon for ϵ>0\epsilon>0 small, under the condition that x0x_0 is a nondegenerate cycle of the linearized unperturbed Hamiltonian system we provide a formula for the distance between any point x0(t)x_0(t) and the trajectories xϵ([0,T])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