Genericity of the multibump dynamics for almost periodic Duffing–like systems

In this paper we consider "slowly" oscillating perturbations of almost periodic Duffing-like systems, i.e., systems of the form $\ddot u=u-(a(t)+\alpha(\omega t))W'(u)$, $t\in\mathbb{R}$, $u\in\mathbb{R}^N$ where $W\in C^{2N}(\mathbb{R}^N,\mathbb{R})$ is superquadratic and $a$ and $\alpha$ are positive and almost periodic. By variational methods, we prove that if $\omega>0$ is small enough then the system admits a multibump dynamics. As a corollary we get that the system $\ddot u=u-a(t)W'(u)$ admits multibump solutions whenever $a$ belongs to an open dense subset of the class of positive almost periodic functions on $\mathbb{R}$.