Generic Singularities of 3D Piecewise Smooth Dynamical Systems
The aim of this paper is to provide a discussion on the current directions of research involving typical singularities of 3D nonsmooth vector fields. A brief survey of known results is also presented.
We describe the dynamical features of a fold–fold singularity in its most basic form and we give a complete and detailed proof of its local structural stability (or instability). In addition, classes of all topological types of a fold–fold singularity are intrinsically characterized. Such proof essentially follows from some lines laid out by Colombo, García, Jeffrey, Teixeira, and others and it offers a rigorous mathematical treatment under clear and crisp assumptions and solid arguments.
One should highlight that the geometric–topological methods employed lead us to the mathematical understanding of the dynamics around a T-singularity. This approach lends itself to applications in generic bifurcation theory. It is worth to say that such subject is still poorly understood in higher dimension.
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