We examine bifurcation phenomena for continuous one-dimensional maps that are piecewise smooth and depend on a parameter μ. In the simplest case, there is a point c at which the map has no derivative (it has two one-sided derivatives). The point c is the border of two intervals in which the map is smooth. As the parameter μ is varied, a fixed point (or periodic point) Eμ may cross the point c, and we may assume that this crossing occurs at μ=0. The investigation of what bifurcations occur at μ=0 reduces to a study of a map fμ depending linearly on μ and two other parameters a and b. A variety of bifurcations occur frequently in such situations. In particular, Eμ may cross the point c, and for μ<0 there can be a fixed point attractor, and for μ>0 there may be a period-3 attractor or even a three-piece chaotic attractor which shrinks to E0 as μ→0. More generally, for every integer m≥2, bifurcations from a fixed point attractor to a period-m attractor, a 2m-piece chaotic attractor, an m-piece chaotic attractor, or a one-piece chaotic attractor can occur for piecewise smooth one-dimensional maps. These bifurcations are called border-collision bifurcations. For almost every point in the region of interest in the (a, b)-space, we state explicitly which border-collision bifurcation actually does occur. We believe this phenomenon will be seen in many applications.
Solitonic Fixed Point Attractors in the Complex Ginzburg–Landau Equation for Associative Memories
