Grazing-sliding bifurcations in planar ℤ₂-symmetric Filippov systems

2025/06/01 by Chen, Xingwu, Fang, Zhihao, Li, Tao
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.01066

Abstract

This paper aims to explore the effect of ℤ₂-symmetry on grazing-sliding bifurcations in planar Filippov systems. We consider the scenario where the unperturbed system is ℤ₂-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 ℤ₂-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.

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