The Boundary-Hopf-Fold Bifurcation in Filippov Systems

This paper studies the codimension-3 boundary-Hopf-fold (BHF) bifurcation of planar Filippov systems. Filippov systems consist of at least one discontinuity boundary locally separating the phase space to disjoint components with different dynamics. Such systems find applications in several fields, for example, mechanical and electrical engineering, and ecology. The BHF bifurcation appears in a subclass of Filippov systems that we call Hopf-transversal systems. In such systems an equilibrium of one vector field goes through a Hopf bifurcation while the other vector field is transversal to the boundary. Depending on the slope of the transversal vector field, different bifurcation scenarios take place. The BHF bifurcation occurs at a critical value of the slope that separates these scenarios. We derive a local normal form for the BHF bifurcation and show the eight different associated bifurcation diagrams. The local 3-parameter normal form topologically models the simplest way to generically unfold the BHF bifurcation. The BHF bifurcation is then studied in a particular example from population dynamics.