Bifurcations and explicit unfoldings of grazing loops connecting one high multiplicity tangent point
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916652012011520 |
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| author | Fang, Zhihao Chen, Xingwu |
| author_facet | Fang, Zhihao Chen, Xingwu |
| contents | For piecewise-smooth differential systems, in this paper we focus on crossing limit cycles and sliding loops bifurcating from a grazing loop connecting one high multiplicity tangent point. For the low multiplicity cases considered in previous publications, the method is to define and analyze return maps following the classic idea of Poincaré. However, high multiplicity leads to that either domains or properties of return maps are unclear under perturbations. To overcome these difficulties, we unfold grazing loops by functional parameters and functional functions, and analyze this unfolding along some specific parameter curve. Relationships between multiplicity and the numbers of crossing limit cycles and sliding loops are given, and our results not only generalize the results obtained in [J. Differential Equations 255(2013), 4403-4436; 269(2020), 11396-11434], but also are new for some specific grazing loops. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_19455 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bifurcations and explicit unfoldings of grazing loops connecting one high multiplicity tangent point Fang, Zhihao Chen, Xingwu Dynamical Systems For piecewise-smooth differential systems, in this paper we focus on crossing limit cycles and sliding loops bifurcating from a grazing loop connecting one high multiplicity tangent point. For the low multiplicity cases considered in previous publications, the method is to define and analyze return maps following the classic idea of Poincaré. However, high multiplicity leads to that either domains or properties of return maps are unclear under perturbations. To overcome these difficulties, we unfold grazing loops by functional parameters and functional functions, and analyze this unfolding along some specific parameter curve. Relationships between multiplicity and the numbers of crossing limit cycles and sliding loops are given, and our results not only generalize the results obtained in [J. Differential Equations 255(2013), 4403-4436; 269(2020), 11396-11434], but also are new for some specific grazing loops. |
| title | Bifurcations and explicit unfoldings of grazing loops connecting one high multiplicity tangent point |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2404.19455 |