Border-Collision Bifurcations in RN

For piecewise-smooth maps, new dynamics can be created by varying parameters such that a fixed point collides with a surface on which the map is nonsmooth. If the map is continuous and has bounded derivatives near but not on the surface, this is referred to as a border-collision bifurcation. Dynamics near border-collision bifurcations are well approximated by piecewise-linear continuous maps. A lack of differentiability allows for substantial complexity and a wide variety of invariant sets can be created in border-collision bifurcations, such as invariant circles and chaotic sets, and several attractors may be created simultaneously. Yet many calculations that would be intractable for smooth nonlinear maps can be performed exactly for piecewise-linear maps and in recent years several new results have been obtained for border-collision bifurcations. This article reviews border-collision bifurcations with a general emphasis on results that apply to maps of any number of dimensions. The paper covers applications, the border-collision normal form, basic properties and dynamics such as fixed points, periodic solutions and mode-locking regions, and more complex phenomena such as multistability, multidimensional attractors, attractors at infinity, homoclinic bifurcations, and unfoldings of codimension-two scenarios.