The Ambrosetti-Prodi periodic problem: Different routes to complex dynamics

We consider a second order nonlinear ordinary differential equation of the form $u^{\mathrm{\prime }\mathrm{\prime }}+f(u)=p(t)$ 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”.