On 3-parameter families of piecewise smooth vector fields in the plane

TL;DR

This paper investigates the local bifurcation structures of three-parameter families of piecewise smooth planar vector fields, focusing on the unfolding of fold-cusp singularities through detailed bifurcation diagrams.

Contribution

It provides a comprehensive analysis of the bifurcation diagrams for three-parameter families of non-smooth vector fields, specifically describing the unfolding of fold-cusp singularities.

Findings

Bifurcation diagrams for three-parameter families are explicitly characterized.

Unfolding of fold-cusp singularities is systematically described.

The study enhances understanding of local dynamics near singularities in piecewise smooth systems.

Abstract

This paper is concerned with the local bifurcation analysis around typical singularities of piecewise smooth planar dynamical systems. Three-parameter families of a class of non$-$smooth vector fields are studied and the tridimensional bifurcation diagrams are exhibited. Our main results describe the unfolding of the so called fold-cusp singularity by means of the variation of 3 parameters.