Fold bifurcation of T-singularities and invariant manifolds in 3D piecewise-smooth dynamical systems

One of the most interesting typical singularity observed in 3D piecewise-smooth dynamical systems is the so called T-singularity (Teixeira Singularity). Throughout this paper, we study a Fold bifurcation involving two T-singularities that, as far as we know, has not yet been fully addressed in the literature. In this bifurcation both T-singularities collide and then disappear when a system parameter is varied. At the collision point appears a type of non-generic T-singularity. For this study we use a normal form and we apply the standard analysis for such class of systems, based on the analysis of the sliding dynamics (Filippov's sliding vector field) and the crossing dynamics (first return map) occurring on the switching boundary. We fully describe the unfolding dynamics of the Fold bifurcation under study and also analyze the existence, stability and bifurcations of invariant manifolds in this scenario. (C) 2019 Elsevier B.V. All rights reserved.