We prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems.

%B Advances in Nonlinear Analysis %I De Gruyter %V 5 %P 367–382 %G eng %R 10.1515/anona-2015-0122