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

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.