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

Abstract

This paper aims to explore the effect of ${\mathbb{Z}}_2$-symmetry on grazing-sliding bifurcations in planar Filippov systems. We consider the scenario where the unperturbed system is ${\mathbb{Z}}_2$-symmetric and its subsystem exhibits a hyperbolic limit cycle grazing the discontinuity boundary at a fold. Employing differential manifold theory, we reveal the intrinsic quantities of unfolding all bifurcations and rigorously demonstrate the emergence of a codimension-two bifurcation under generic ${\mathbb{Z}}_2$-symmetric perturbations within the Filippov framework. After deriving an explicit non-degenerate condition with respect to parameters, we systematically establish the complete bifurcation diagram with exact asymptotics for all bifurcation boundaries by displacement map method combined with asymptotic analysis.