  We consider a second order nonlinear ordinary differential equation of the
form $u'' + f(u) = p(t)$ where the forcing term $p(t)$ is a $T$-periodic
function and the nonlinearity $f(u)$ satisfies the properties of
Ambrosetti-Prodi problems. We discuss the existence of infinitely many periodic
solutions as well as the presence of complex dynamics under different
conditions on $p(t)$ and by using different kinds of approaches. On the one
hand, we exploit the Melnikov's method and, on the other hand, the concept of
"topological horseshoe".
