Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Zhihao Fang, Xingwu Chen

TL;DR

The paper addresses bifurcations of crossing limit cycles $L_c$ and sliding loops $L_s$ arising from a grazing loop $L^{gra}_*$ that connects a tangent point of multiplicity $$(1,m)$$ in a piecewise-smooth system. It introduces an unfolding framework using a scalar parameter $\alpha$ and a smooth function $\psi(x,\boldsymbol{k})$, constructed from smooth cutoffs, to render the grazing scenario perturbable and analyzable along a chosen parameter curve. The main result, Theorem $['thm1']$, provides lower bounds on the total number of bifurcating $L_c$ and $L_s$ ($\beta_c+\beta_s$) as functions of $m$ and the grazing-type (e.g., $S$-$I$, $S$-$V$, $S$-$L$, $S$-$R$), generalizing prior work to general multiplicities and yielding new outcomes for certain grazing loops. The method clarifies the role of high multiplicity in the bifurcation structure of piecewise-smooth systems and offers a constructive approach to realize multiple bifurcations through carefully chosen perturbations, with implications for dynamics governed by Filippov-type switching on $\Sigma$.

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.

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

TL;DR

The paper addresses bifurcations of crossing limit cycles and sliding loops arising from a grazing loop that connects a tangent point of multiplicity in a piecewise-smooth system. It introduces an unfolding framework using a scalar parameter and a smooth function , constructed from smooth cutoffs, to render the grazing scenario perturbable and analyzable along a chosen parameter curve. The main result, Theorem , provides lower bounds on the total number of bifurcating and () as functions of and the grazing-type (e.g., -, -, -, -), generalizing prior work to general multiplicities and yielding new outcomes for certain grazing loops. The method clarifies the role of high multiplicity in the bifurcation structure of piecewise-smooth systems and offers a constructive approach to realize multiple bifurcations through carefully chosen perturbations, with implications for dynamics governed by Filippov-type switching on .

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.
Paper Structure (4 sections, 4 theorems, 58 equations, 8 figures)

This paper contains 4 sections, 4 theorems, 58 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminary lemmas
  3. Proof of Theorem \ref{['thm1']}
  4. Conclusions and method discussions

Key Result

Theorem 1.1

Assume that $f^+(0,0)f^-(0,0)>0$ in system pws2. Then there exists a perturbation system of form such that the following statements hold. Here $\psi:=\psi(x,{\boldsymbol k})$ is a $C^{\infty}$ function satisfying $\psi(x,{\boldsymbol 0})\equiv 0$, $\dot\psi:=d\psi/dx$, $\alpha\in\mathbb{R},{\boldsymbol \lambda}:=(\lambda_1,...,\lambda_m)\in\mathbb{R}^m,~{\boldsymbol k}:=(k_1,...,k_n)\in\mathbb{R

Figures (8)

  • Figure 1: visibility of tangent point
  • Figure 2: examples of $L^{cri},~L^{cro},~L^{gra}$
  • Figure 3: grazing loop $L^{gra}_*$
  • Figure 4: $\psi(x,{\boldsymbol k})$ with $d=3$ and ${\boldsymbol k}\in{\cal K}, k_8,k_9,k_{10}<0$
  • Figure 5: transition map $V(r)$
  • ...and 3 more figures

Theorems & Definitions (6)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof : Proof of Theorem \ref{['thm1']}(a)
  • proof : Proof of Theorem \ref{['thm1']}(b)