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 Z0 in 3D presenting a T-singularity is considered and a complete picture of its dynamics is obtained in the following way: (i) Z0 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); (ii) All trajectories of Z0 converge to the surface $$\pi_0$$; (iii) given an arbitrary integer $$k \geq 0$$ then Z0 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; (iv) the origin can be chosen as an asymptotic stable equilibrium of $$Z_{\varepsilon}$$ when k = 0; and finally, (v) Z0 has infinite codimension in the set of all 3D piecewise smooth vector fields.