Grazing-sliding bifurcations in planar $\mathbb{Z}_2$-symmetric Filippov systems
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909848940052480 |
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| author | Chen, Xingwu Fang, Zhihao Li, Tao |
| author_facet | Chen, Xingwu Fang, Zhihao Li, Tao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01066 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Grazing-sliding bifurcations in planar $\mathbb{Z}_2$-symmetric Filippov systems Chen, Xingwu Fang, Zhihao Li, Tao Dynamical Systems 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. |
| title | Grazing-sliding bifurcations in planar $\mathbb{Z}_2$-symmetric Filippov systems |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2506.01066 |