On the local structure of degenerate Teixeira singularities in 3D Filippov systems

The main goal of this paper is to emphasize the richness of the dynamics emanating from degeneracies at the so-called T-singularity in $3D$ Filippov systems. More specifically, we characterize the local sliding and crossing dynamics around an invisible two-fold singularity in $3D$ Filippov systems which presents a degeneracy arising from the contact between the tangency fold curves of a system with the switching manifold. In particular, we prove that, when the contact between such curves is 2 or 3 at this point, then it presents a nonsmooth diabolo emanating from it which has one branch or two branches, respectively.

We also analyze global bifurcations of a family of Filippov systems which presents an invisible two-fold singularity having a cubic contact between the fold curves and we show that there is an invariant surface foliated by crossing heteroclinic orbits between $T$-singularities bifurcating from this degenerate singularity. Finally, we show that such kind of scenario appears naturally in applied models of switched electronic circuits and this singularity is realized in an specific model.