Bifurcation and chaos near sliding homoclinics (Q963712)

The paper is devoted to the study of the chaotic behaviour of a time 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. It is assumed that the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations. Following a functional analytic approach, a Melnikov-like function \(\mathcal M(\alpha)\) is constructed in such a way that if \(\mathcal M(\alpha)\) has a simple zero at some point, then the system has solutions that behave chaotically. Applications of this result to almost periodic, quasi-periodic and periodic cases are given.