Bifurcations at a degenerate two-fold singularity and crossing limit cycles

Abstract

In the 1990s Teixeira studied discontinuous piecewise-smooth (DPWS) dynamical systems in R3, whose state space is divided in two open regions by a plane acting as the switching manifold, and presenting two lines of quadratic tangency, one for each involved vector field. We call T-singularity (Teixeira singularity) the point of transverse intersection of the quadratic tangency lines that are of the invisible type. In this paper, we analyze DPWS systems that present two branches of T-singularities. In our approach we consider the cases where either a pseudo-equilibrium collides to just one of the two T-singularities or the two T-singularities (and also a pseudo-equilibrium between them) collides in a single point. As a consequence of these bifurcations, until now not covered by the literature, we can observe the birth of limit cycles, determine their stability and provide an upper limit for the number of limit cycles that occur in this scenario.