Sliding bifurcations of periodically perturbed Filippov systems

In this paper, we study sliding bifurcations of periodically perturbed Filippov systems, instead of autonomous perturbations considered in most publications. By defining different smooth Poincaré maps or equivalently displacement maps for standard, sliding, and crossing periodic solutions, respectively, we overcome the difficulty of lacking a unified smooth Poincaré map to capture all possible types of periodic solutions in sliding bifurcations, and finally develop new criteria to identify the existence and number of periodic solutions in four different scenarios of sliding bifurcations. Our work not only extends previous works from non-degenerate unperturbed periodic solution to degenerate one, but also provides an understanding of the well-known codimension-one sliding bifurcations from a non-autonomous perspective. We also show applications of the obtained theoretical results to periodic perturbations of some polynomial Filippov systems, including models in the fields of classical mechanics.