We prove the existence of multiple periodic solutions for a planar Hamiltonian system generated from the second order scalar ODE of Duffing type with g satisfying a one-sided condition of sublinear type. We consider the classical approach based on the Poincaré-Birkhoff fixed point theorem as well as some refinements on the side of the theory of bend-twist maps and topological horseshoes. We focus our analysis to the case of a stepwise weight function, in order to highlight the underlying geometrical structure.
