Bifurcations of degenerate homoclinic solutions in discontinuous systems under non-autonomous perturbations

The main aim of this paper is to study bifurcations of bounded solutions from a degenerate homoclinic solution for discontinuous systems under non-autonomous perturbations. We use Lyapunov-Schmidt reduction to give bifurcation equations and prove that a single parameter is enough to unfold two distinct homoclinic solutions bifurcated from the unperturbed degenerate homoclinic solution. Furthermore, we give an example of a periodically perturbed piecewise smooth differential equation in R 4 to support our conclusions.