Border-collision bifurcations including “period two to period three” for piecewise smooth systems

Abstract We examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter μ. In the simplest case there is a surface Γ in phase space along which the map has no derivative (or has two one-sided derivatives). Γ is the border of two regions in which the map is smooth. As the parameter μ is varied, a fixed point Eμ may collide with the border Γ, and we may assume that this collision occurs at μ = 0. A variety of bifurcations occur frequently in such situations, but never or almost never occur in smooth systems. In particular Eμ may cross the border and so will exist for μ 0 but it may be a saddle in one case, say μ 0. For μ 0 there may be a stable period 3 orbit which similarly shrinks to E0 as μ→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.