Pseudo-Bautin bifurcation for a non-generic family of 3D Filippov systems

We consider the non-generic family of 3D piecewise linear systems, with a discontinuity plane that have two parallel tangency lines, such that the region between them is the sliding region. It is known that the change of stability of the sliding region gives rise to the called pseudo-Hopf bifurcation. The stability of the crossing limit cycle that emerges from this bifurcation mechanism is characterized by two control parameters. In this document we consider one of these control parameters as a bifurcation parameter and establish the existence of a curve of saddle-node bifurcation points for crossing limit cycles. When we put together this two bifurcation mechanisms in a two-parametric unfolding, we obtain the called pseudo-Bautin bifurcation, because the local geometry of the bifurcation curves in the bifurcation diagram is the same as the Bautin bifurcation for smooth dynamical systems. Finally, we apply this result to state feedback control systems.