On the dynamics of non-autonomous systems in a neighborhood of a homoclinic trajectory

This article is devoted to the study of a 2-dimensional piecewise smooth (but possibly) discontinuous dynamical system, subject to a non-autonomous perturbation; we assume that the unperturbed system admits a homoclinic trajectory ⃗γ(t). Our aim is to analyze the dynamics in a neighborhood of ⃗γ(t) as the perturbation is turned on, by defining a Poincar´e map and evaluating fly time and space displacement of trajectories performing a loop close to ⃗γ(t). Besides their intrinsic mathematical interest, these results can be thought of as a first step in the analysis of several interesting problems, such as the stability of a homoclinic trajectory of a non-autonomous ODE and a possible extension of Melnikov chaos to a discontinuous setting.