Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincar\'{e}-Birkhoff approach

In this paper we prove the existence of multiple periodic solutions (harmonic and subharmonic) for a class of planar Hamiltonian systems which include the case of the second order scalar ODE $x^{\mathrm{\prime }\mathrm{\prime }}+a(t)g(x)=0$ with $g$ satisfying a one-sided condition of sublinear type. We consider the classical approach based on the Poincar'{e}-Birkhoff fixed point theorem as well as some refinements on the side of the theory of bend-twist maps and topological horseshoes. The case of complex dynamics is investigated, too.