Pseudo-Cusp Bifurcation in a Class of 3D Piecewise Linear Differential Systems

Consider the nongeneric family of 3D piecewise linear differential systems, with a discontinuity plane that has two parallel invisible 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 a Crossing Limit Cycle (CLC). The stability of the CLC that emerges from this bifurcation mechanism is characterized by two control parameters, which are considered as bifurcation parameters. One of them is associated with the existence of saddle-node bifurcation points for CLCs, while the other one is associated with the existence of cusp bifurcation points for CLCs. By putting together these three bifurcation mechanisms in a three-parametric unfolding, we obtain a codimension-3 bifurcation point, which we will call the pseudo-cusp bifurcation. The 3D bifurcation diagram is characterized by regions with zero, one, two and three CLCs.