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Border-collision bifurcations for piecewise smooth one-dimensional maps. (English) Zbl 0885.58060

Summary: We examine bifurcation phenomena for continuous one-dimensional maps that are piecewise smooth and depend on a parameter \(\mu\). 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 \(\mu\) is varied, a fixed point (or periodic point) \(E_\mu\) may cross the point \(c\) and we may assume that this crossing occurs at \(\mu =0\). The investigation of what bifurcations occur at \(\mu=0\) reduces to a study of a map \(f_\mu\) depending linearly on \(\mu\) and two other parameters \(a\) and \(b\). A variety of bifurcations occur frequently in such situations. In particular, \(E_\mu\) may cross the point \(c\), and for \(\mu<0\) there can be a fixed point attractor, and for \(\mu>0\) there may be a period-3 attractor or even a three-piece chaotic attractor which shrinks to \(E_0\) as \(\mu \to 0\). More generally, for every integer \(m\geq 2\), 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.

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
37E99 Low-dimensional dynamical systems
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