# Generic Singularities of 3D Piecewise Smooth Dynamical Systems

## TL;DR

This paper analyzes the local dynamics and stability of fold-fold singularities in 3D nonsmooth vector fields, providing rigorous proofs and a comprehensive topological classification, advancing understanding in nonsmooth dynamical systems.

## Contribution

It offers a detailed mathematical analysis and classification of fold-fold singularities in 3D nonsmooth systems, including stability results and topological types.

## Key findings

- Complete proof of local structural stability/instability of fold-fold singularities.
- Intrinsic topological classification of all fold-fold singularity types.
- Mathematical framework applicable to bifurcation analysis in nonsmooth systems.

## Abstract

The aim of this paper is to provide a discussion on current directions of research involving typical singularities of 3D nonsmooth vector fields. A brief survey of known results is presented. The main purpose of this work is to describe the dynamical features of a fold-fold singularity in its most basic form and to 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 firstly from some lines laid out by Colombo, Garc\'ia, Jeffrey, Teixeira and others and secondly offers a rigorous mathematical treatment under clear and crisp assumptions and solid arguments. One should to highlight that the geometric-topological methods employed lead us to the completely 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.