Stability conditions in piecewise smooth dynamical systems at a two-fold singularity (Q1936290)

The authors consider vector fields expressible in the form \(\dot{q}=Z(q)\), where \(q\in \mathbb{R}^3\) is a state vector and \(Z=(X,Y)\) is a piecewise smooth map with the aim to study their dynamics near typical singularity and especially the solutions stability. The discontinuities concentrate on a codimension-one submanifold \(\Sigma\) of \(\mathbb{R}^3\) (switching manifold). Solution orbits on \(\Sigma\), when they exist, are defined according to Filippov's convention [\textit{A. F. Filippov}, Differential equations with discontinuous right-hand sides. Dordrecht etc.: Kluwer Academic Publishers (1988; Zbl 0664.34001)]. In contrast to the 2D-case, in 3D non-smooth vector fields the interesting object -- the two-fold singularly -- arises which is studies in this article.