Versal unfoldings of the boundary equilibria in two-dimensional continuous piecewise linear systems

The aim of this paper is to study versal unfoldings of boundary equilibria of planar continuous piecewise linear systems with a switching line, since the boundary equilibria of these systems are structurally unstable under perturbation with the same switching line. We present bifurcation diagrams and phase portraits of these versal unfoldings, except those of the boundary-monodromy equilibria (which were described in [2]). We show that the codimension of bifurcations near the boundary equilibria of planar continuous piecewise linear systems with the switching line is at most 4, and demonstrate that at most one limit cycle can be perturbed in a small neighborhood of boundary equilibria.