On border-collision bifurcations in a pulse system

Considering a piecewise smooth map describing the behavior of a pulse-modulated control system, we discuss border-collision related phenomena. We show that in the parameter space which corresponds to the domain of oscillatory mode a mapping is piecewise linear continuous. It is well known that in piecewise linear maps, classical bifurcations, for example, period doubling, tangent, fold bifurcations become degenerate (“degenerate bifurcations”), combining the properties of both smooth and border-collision bifurcations. We found unusual properties of this map, that consist in the fact that border-collision bifurcations of codimension one, including degenerate ones, occur when a pair of points of a periodic orbit simultaneously collides with two switching manifolds. This paper also discuss bifurcations of chaotic attractors such as merging and expansion (“interior”) crises, associated with homoclinic bifurcations of unstable periodic orbits.