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

TL;DR

This paper studies bifurcations of grazing loops in piecewise-smooth systems with high multiplicity tangent points, introducing an unfolding method to analyze complex behaviors and generalize previous results.

Contribution

It develops a novel unfolding approach for high multiplicity grazing loops, overcoming limitations of traditional return map analysis and extending bifurcation theory.

Findings

Relationships between multiplicity and bifurcation outcomes are established.

Unfolding parameters reveal new bifurcation scenarios for grazing loops.

Results generalize previous low multiplicity cases and introduce new phenomena.

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 Poincar\'e. 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.