Attractivity, degeneracy and codimension of a typical singularity in 3D piecewise smooth vector fields

Abstract

We address the problem of understanding the dynamics around typical singular points of $3D$ piecewise smooth vector fields. A model $Z_0$ in $3D$ presenting a T-singularity is considered and a complete picture of its dynamics is obtained in the following way: \textit{(i)} $Z_0$ has an invariant plane $\pi_0$ filled up with periodic orbits (this means that the restriction $Z_0 |_{\pi_0}$ is a center around the singularity), \textit{(ii)} All trajectories of $Z_0$ converge to the surface $\pi_0$, and such attraction occurs in a very non-usual and amazing way, \textit{(iii)} given an arbitrary integer $k\geq 0$ then $Z_0$ can be approximated by $\pi_0$-invariant piecewise smooth vector fields $Z_{\varepsilon}$ such that the restriction $Z_{\varepsilon} |_{\pi_0}$ has exactly $k$-hyperbolic limit cycles, \textit{(iv)} the origin can be chosen as an asymptotic stable equilibrium of $Z_{\varepsilon}$ when $k=0$, and finally, \textit{(v)} $Z_0$ has infinite codimension in the set of all $3D$ piecewise smooth vector fields.