Codimension-two singularities on the stability boundary in 2D Filippov systems

Bifurcation theory provides powerful tools for the analysis of the dynamics of open-loop or closed-loop nonlinear control systems. These systems are Filippov (or piecewise smooth) when the dynamics depends discontinuously on the state, for example as a consequence relay feedback actions. In this paper we contribute to the analysis of codimension-two bifurcations in Filippov systems by reporting some results on the equilibrium bifurcations of 2D systems that involve a sliding limit cycle. There are only two such local bifurcations: a degenerate boundary focus that we call homoclinic boundary focus ; and the boundary Hopf. We address both of them, and provide the complete set of curves that exist around such codimension-two bifurcation points. Existing numerical software can be used to exploit these results for the analysis of the stability boundaries of nonlinear piecewise smooth control systems. In the final part of this paper, we discuss a 2D Filippov system modelling an ecosystem subject to on-off harvesting control that exhibits both codimension-two bifurcations.