Multibump solutions for Duffing–like systems

We study the problem of existence of homoclinic solutions of a second order asymptotically periodic Hamiltonian system: {\sl find $q\in C^2(\mathbb{R},\mathbb{R}^N)\setminus\{0\}$ such that:} $$ \ddot q=q-V'(t,q)\ ,\quad q(t)\to 0\ \ {\sl and}\ \ \dot q(t)\to 0\ \ {\sl as}\ \ t\to\pm\infty\, \eqno({\rm HS}) $$ where it is assumed that the origin is a local maximum for the corresponding potential, uniformly in time, and that $V'$ is asymptotic, as $t\to\pm\infty$, to time periodic and superquadratic functions $V'_\pm$. We prove via variational methods that if the stable and unstable manifolds associated to the origin of one of the systems at infinity have countable intersection then the problem (HS) has infinitely many homoclinic solutions of multibump type.