BORDER-COLLISION BIFURCATIONS INCLUDING PERIOD 2 TO PERIOD 3 FOR PIECEWISE SMOOTH SYSTEMS

We examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter mu. In the simplest case there is a surface GAMMA in phase space along which the map has no derivative (or has two one-sided derivatives). GAMMA is the border of two regions in which the map is smooth. As the parameter mu is varied, a fixed point E(mu) may collide with the border GAMMA, and we may assume that this collision occurs at mu = 0. A variety of bifurcations occur frequently in such situations, but never or almost never occur in smooth systems. In particular E(mu) may cross the border and so will exist for mu < 0 and for mu > 0 but it may be a saddle in one case, say mu < 0, and it may be a repellor for mu > 0. For mu < 0 there can be a stable period two orbit which shrinks to the point E0 as mu --> 0, and for mu > 0 there may be a stable period 3 orbit which similarly shrinks to E0 as mu --> 0. Hence one observes the following stable periodic orbits: a stable period 2 orbit collapses to a point and is reborn as a stable period 3 orbit. We also see analogously "stable period 2 to stable period p orbit bifurcations", with p = 5,11,52, or period 2 to quasi-periodic or even to a chaotic attractor. We believe this phenomenon will be seen in many applications.