Bifurcation and chaos near sliding homoclinics

We study the chaotic behaviour of a little dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface. We assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations. Following a functional analytic approach we construct a Melnikov-like function M(alpha) ill Such a way that if M(alpha) has a simple zero at some point, then the system has solutions that behave chaotically. Applications of this result to quasi-periodic systems are also given. (C) 2009 Elsevier Inc. All rights reserved.