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

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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 π₀ filled up with periodic orbits (this means that the restriction Z₀ |π₀ is a center around the singularity), (ii) All trajectories of Z₀ converge to the surface π₀, and such attraction occurs in a very non-usual and amazing way, (iii) given an arbitrary integer k≥ 0 then Z₀ can be approximated by π₀-invariant piecewise smooth vector fields Zε such that the restriction Zε |π₀ has exactly k-hyperbolic limit cycles, (iv) the origin can be chosen as an asymptotic stable equilibrium of Zε when k=0, and finally, (v) Z₀ has infinite codimension in the set of all 3D piecewise smooth vector fields.