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

TL;DR

This paper investigates how $ ext{Z}_2$-symmetry influences grazing-sliding bifurcations in planar Filippov systems, revealing a codimension-two bifurcation and providing a detailed bifurcation diagram with asymptotic analysis.

Contribution

It introduces a rigorous analysis of $ ext{Z}_2$-symmetric effects on bifurcations in Filippov systems, identifying a codimension-two bifurcation and deriving explicit conditions for bifurcation boundaries.

Findings

Identification of a codimension-two bifurcation under symmetric perturbations

Explicit non-degenerate bifurcation conditions derived

Complete bifurcation diagram with asymptotic boundary descriptions

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.