We prove the existence of periodic solutions for a planar non-autonomous Hamiltonian system which is a small perturbation of an autonomous system, in the presence of a non-isochronous period annulus. To this aim we use the Poincar\'e-Birkhoff fixed point theorem, even if the boundaries of the annulus are neither assumed to be invariant for the Poincar\'e map, nor to be star-shaped. As a consequence, we show how to deal with the problem of bifurcation of subharmonic solutions near a given nondegenerate periodic solution. In this framework, we only need little regularity assumptions, and we do not need to introduce any Melnikov type functions.

Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré-Birkhoff theorem

Sabatini, Marco;
2012-01-01

Abstract

We prove the existence of periodic solutions for a planar non-autonomous Hamiltonian system which is a small perturbation of an autonomous system, in the presence of a non-isochronous period annulus. To this aim we use the Poincar\'e-Birkhoff fixed point theorem, even if the boundaries of the annulus are neither assumed to be invariant for the Poincar\'e map, nor to be star-shaped. As a consequence, we show how to deal with the problem of bifurcation of subharmonic solutions near a given nondegenerate periodic solution. In this framework, we only need little regularity assumptions, and we do not need to introduce any Melnikov type functions.
2012
1
A., Fonda; Sabatini, Marco; F., Zanolin
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/88464
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