Two Degenerate Boundary Equilibrium Bifurcations in Planar Filippov Systems

Abstract

We contribute to the analysis of codimension-two bifurcations in discontinuous systems by studying all equilibrium bifurcations of 2D Filippov systems that involve a sliding limit cycle. There are only two such local bifurcations: (1) a degenerate boundary focus, which we call the homoclinic boundary focus (HBF), and (2) the boundary Hopf (BH). We prove that—besides local bifurcations of equilibria and pseudoequilibria—the universal unfolding of the HBF singularity includes a codimension-one global bifurcation at which a sliding homoclinic orbit to a pseudosaddle exists, while that of the BH singularity has a codimension-one bifurcation curve along which a cycle grazing occurs. We define two canonical forms, one for each singularity, to which a generic 2D Filippov system can be locally reduced by smooth changes of variables and parameters and time reparametrization. Explicit genericity conditions are also provided, as well as the asymptotics of the bifurcation curves in the two-parameter space. We show that both studied codimension-two bifurcations occur in a known 2D Filippov system modeling an ecosystem subject to on-off harvesting control, and we provide two Mathematica scripts that automatize all computations.