Crossing-sliding bifurcations in planar $\mathbb{Z}_2$-symmetric Filippov systems

In this paper we investigate the crossing-sliding bifurcations of planar Filippov systems with $\mathbb{Z}_2$-symmetry. Such bifurcations are triggered by the perturbations of a critical crossing cycle and constitute an important class of discontinuity-induced bifurcations. By constructing transition maps and developing a decomposition theorem of functions to overcome the difficulty of describing bifurcation boundaries in multi-parameter settings, we systematically characterize the codimension-one and codimension-two bifurcation scenarios through the explicit statement of non-degenerate conditions and the presentation of the corresponding bifurcation diagrams. The asymptotic properties of all bifurcation curves are also derived.