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

TL;DR

This paper analyzes the dynamics near a typical singularity in 3D piecewise smooth vector fields, revealing invariant structures, convergence behaviors, and the ability to approximate these fields with systems exhibiting various numbers of limit cycles.

Contribution

It provides a complete description of the dynamics around a T-singularity in 3D piecewise smooth vector fields and introduces the concept of infinite codimension for such singularities.

Findings

Invariant plane filled with periodic orbits.

Trajectories converge to the invariant plane.

Ability to approximate with systems having any number of hyperbolic limit cycles.

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.