Boundary Hopf bifurcations in three-dimensional Filippov systems

Abstract

For piecewise-smooth ordinary differential equations, the occurrence of a Hopf bifurcation on a switching surface is known as a boundary Hopf bifurcation. Boundary Hopf bifurcations are codimension-two, so occur at points in two-parameter bifurcation diagrams. From any such point there issues a curve of grazing bifurcations, where the limit cycle born in the Hopf bifurcation hits the switching surface. For Filippov systems, these are usually grazing-sliding bifurcations whose local dynamics are dictated by piecewise-linear maps. In general, these maps have many independent parameters and extraordinarily rich dynamical behaviour. We show that for three-dimensional Filippov systems only a two-parameter family of piecewise-linear maps is relevant, because sliding motion induces a loss of dimension, and the stability of the limit cycle is degenerate at the Hopf bifurcation. We derive explicit formulas for the two parameters in terms of quantities associated with the boundary Hopf bifurcation, and perform a comprehensive numerical analysis to characterise the attractor of the family, which may be chaotic. The results are illustrated with a pedagogical example, a pest control model, and a model of a food chain with threshold-based harvesting. To evaluate the parameters, we use a formula for the linear term of the discontinuity map associated with grazing-sliding bifurcations. In this paper we present a new, simpler derivation of this formula for $n$-dimensional systems based on displacements from a virtual counterpart.