Canonical forms of 3d cusp-fold singularities and its unfoldings

TL;DR

This paper classifies 32 topologically distinct canonical forms of 3D cusp-fold singularities in piecewise smooth vector fields, analyzing their bifurcations and topological features.

Contribution

It introduces a complete classification of 3D cusp-fold singularities and their unfoldings, providing a comprehensive topological framework for these complex singularities.

Findings

32 canonical forms identified

Topological distinctions based on tangencies and regions

Bifurcation analysis of canonical forms

Abstract

In this paper we obtain 32 canonical forms for 3D piecewise smooth vector fields presenting the so called cusp-fold singularity. All these canonical forms are topologically distinct and collect the main topological aspects of the singularities described as kind of the tangencies involved and positions of the sliding, escaping and crossing regions. Also, one-parameter bifurcations of these canonical forms are presented and the topologically equivalent piecewise smooth vector fields are obtained.