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Grazing-sliding bifurcations in planar $\mathbb{Z}_2$-symmetric Filippov systems

Xingwu Chen, Zhihao Fang, Tao Li

TL;DR

The paper analyzes grazing-sliding bifurcations in planar Filippov systems endowed with $\mathbb{Z}_2$-symmetry, focusing on a hyperbolic limit cycle that grazes the discontinuity at a fold to form a symmetric figure-eight. By employing differential-geometry tools and an explicit two-parameter unfolding, it proves that the grazing-sliding bifurcation set forms a codimension-two submanifold and derives a nondegeneracy condition that yields a complete, asymptotically described bifurcation diagram via a displacement-map approach. The authors construct a rigorous local framework using two preliminary lemmas, then determine, through a sequence of lemmas and a detailed expansion of transition maps, how nine distinct dynamical regimes arise and transition as parameters vary. The results are validated with an example that demonstrates the codimension-two unfolding in a concrete $\mathbb{Z}_2$-symmetric Filippov system, highlighting the practical relevance of the diagrammatic scenarios for sliding, grazing, and crossing cycles.

Abstract

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.

Grazing-sliding bifurcations in planar $\mathbb{Z}_2$-symmetric Filippov systems

TL;DR

The paper analyzes grazing-sliding bifurcations in planar Filippov systems endowed with -symmetry, focusing on a hyperbolic limit cycle that grazes the discontinuity at a fold to form a symmetric figure-eight. By employing differential-geometry tools and an explicit two-parameter unfolding, it proves that the grazing-sliding bifurcation set forms a codimension-two submanifold and derives a nondegeneracy condition that yields a complete, asymptotically described bifurcation diagram via a displacement-map approach. The authors construct a rigorous local framework using two preliminary lemmas, then determine, through a sequence of lemmas and a detailed expansion of transition maps, how nine distinct dynamical regimes arise and transition as parameters vary. The results are validated with an example that demonstrates the codimension-two unfolding in a concrete -symmetric Filippov system, highlighting the practical relevance of the diagrammatic scenarios for sliding, grazing, and crossing cycles.

Abstract

This paper aims to explore the effect of -symmetry on grazing-sliding bifurcations in planar Filippov systems. We consider the scenario where the unperturbed system is -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 -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.

Paper Structure

This paper contains 7 sections, 14 theorems, 87 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Notions
  3. Main results
  4. Preliminary lemmas
  5. Proof of Theorem \ref{['thm-codim']}
  6. Proof of Theorem \ref{['mainthm']}
  7. Example

Key Result

Theorem 3.1

Assume that $Z_0\in\Omega_0$ has a figure eight loop $\Upsilon_0$ characterized by (H1) and (H2). Then for a sufficiently small figure eight annulus neighborhood $\mathcal{A}$ of $\Upsilon_0$ there exists a neighborhood $\mathcal{U}\subset\Omega$ of $Z_0$ such that $\mathcal{U}_0$ is a codimension-

Figures (10)

  • Figure 3: Grazing-sliding bifurcation in $\mathbb{Z}_2$-symmetric Filippov systems except regions $R_1, R_2$.
  • Figure 4: Bifurcations in the regions $R_1$ and $R_2$.
  • Figure : $\mu<0$
  • Figure : $\mu<0$
  • Figure : $\mu<0$
  • ...and 5 more figures

Theorems & Definitions (25)

  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 5.1
  • proof
  • proof : Proof of Theorem \ref{['thm-codim']}
  • Lemma 6.1
  • ...and 15 more