Bifurcations and explicit unfoldings of grazing loops connecting one high multiplicity tangent point

2024/04/30 by Zhihao Fang, Xingwu Chen, Fang, Zhihao +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Advanced Differential Geometry Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2404.19455

openalex publication_date 2024/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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