THE AMBROSETTI-PRODI PERIODIC PROBLEM: DIFFERENT ROUTES TO COMPLEX DYNAMICS

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 properties related to problems of Ambrosetti-Prodi type. 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”.