On the chaotic behavior of non-flat billiards

We study the problem of the motion of a particle on a non-flat billiard. The particle is subject to the gravity and to a small amplitude periodic (or almost periodic) forcing and is reflected with respect to the normal axis when it hits the boundary of the billiard. We prove that the unperturbed problem has an impact homoclinic orbit and give a Melnikov type condition so that the perturbed problem exhibit chaotic behavior in the sense of Smale's horseshoe.