Degenerate grazing-sliding bifurcations in planar Filippov systems

Abstract

In this paper we contribute to a class of degenerate grazing-sliding bifurcations in planar Filippov systems, assuming that the unperturbed system has a generalized homoclinic loop to a fold-fold point. Depending on the type of this fold-fold point, we exhibit some codimension-2 bifurcation diagrams as well as the asymptotics of all bifurcation curves. Our work not only reports several complete unfoldings in the investigation of degenerate grazing-sliding bifurcations, but also answers the open problem about the maximum number of limit cycles bifurcating from the generalized homoclinic loop for the case of fold-fold presented in Liang et al. (2013) [30].