  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 $x'' + a(t)g(x) = 0$ 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.
