Grazing bifurcations in periodic hybrid systems

Grazing bifurcations occur when a small parameter variation induces a change in the event sequence of a hybrid system, i.e., a system where continuous dynamics and discrete events strongly interact. At such a bifurcation, the system trajectory makes tangential contact with (grazes) an event triggering hypersurface. This bounding case separates regions of (generally) quite different dynamic behaviour. The paper formulates the conditions governing grazing bifurcation points, and extends those conditions to limit cycles. A shooting method is used to solve for bifurcating limit cycles. The approach is applicable for general nonlinear hybrid systems.