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

Alessandro Fonda, Marco Sabatini, Fabio Zanolin


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 Poincaré-Birkhoff fixed point theorem, even if
the boundaries of the annulus are neither assumed to be invariant for the Poincaré 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; Poincaré-Birkhoff; nonlinear dynamics

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