Corner collision and grazing sliding: practical examples of border collision bifurcations

This chapter gives an overview of the main types of nonsmooth transitions which can be observed in piecewise smooth dynamical systems. Particular attention is given to those events involving interactions with the discontinuity boundary of fixed points of piecewise-smooth maps and limit cycles of piecewise-smooth flows. Strategies to classify these phenomena are discussed. It is shown that only few cases lead to maps which are locally piecewise linear to leading order. A nonlinear friction oscillator is used as a representative example to illustrate the main ideas introduced in the chapter.