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

TL;DR

This paper systematically analyzes crossing-sliding bifurcations in symmetric planar Filippov systems, providing explicit conditions, bifurcation diagrams, and asymptotic properties to understand these complex discontinuity-induced phenomena.

Contribution

It introduces a decomposition theorem and transition maps to explicitly characterize bifurcation scenarios in symmetric Filippov systems, advancing the understanding of their bifurcation structure.

Findings

Explicit bifurcation diagrams derived for various scenarios

Asymptotic behavior of bifurcation curves characterized

Systematic classification of codimension-one and two bifurcations

Abstract

In this paper we investigate the crossing-sliding bifurcations of planar Filippov systems with $\mathbb{Z}_2$-symmetry. Such bifurcations are triggered by the perturbations of a critical crossing cycle and constitute an important class of discontinuity-induced bifurcations. By constructing transition maps and developing a decomposition theorem of functions to overcome the difficulty of describing bifurcation boundaries in multi-parameter settings, we systematically characterize the codimension-one and codimension-two bifurcation scenarios through the explicit statement of non-degenerate conditions and the presentation of the corresponding bifurcation diagrams. The asymptotic properties of all bifurcation curves are also derived.