Stability and cyclicity of polycycles in non-smooth planar vector fields

TL;DR

This paper extends stability and bifurcation results for polycycles from smooth to non-smooth planar vector fields, analyzing their stability and limit cycle bifurcations in Filippov systems.

Contribution

It generalizes existing smooth vector field results to non-smooth systems, including complex singularities, and characterizes stability and bifurcation scenarios.

Findings

Polycycles' stability criteria are established for non-smooth systems.

Bifurcation of at most one limit cycle under certain conditions is proved.

At least one limit cycle emerges for each singularity in other conditions.

Abstract

In this paper we extend three results about polycycles (also known as graphs) of planar smooth vector field to planar non-smooth vector fields (also known as piecewise vector fields, or Filippov systems). The polycycles considered here may contain hyperbolic saddles, semi-hyperbolic saddles, saddle-nodes and tangential singularities of any degree. We determine when the polycycle is stable or unstable. We prove the bifurcation of at most one limit cycle in some conditions and at least one limit cycle for each singularity in other conditions.