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arXiv · 2506.01066

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

Xingwu Chen·Zhihao Fang·Tao Li
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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.

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Xingwu Chen, Zhihao Fang, Tao Li. 2025-06-01. Grazing-sliding bifurcations in planar $\mathbb{Z}_2$-symmetric Filippov systems. https://arxiv.org/abs/2506.01066

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