TABLE OF CONTENTS The above examples have illustrated some of the peculiar bifurcation phenomena that arise in piecewise linear maps. For piecewise-smooth dynamical systems one can in general distinguish between two types of bifurcations. The first type is similar to the bifurcations we know for smooth dynamical systems. These include the local bifurcations (saddle-node, pitchfork, period-doubling, and Hopf bifurcations), where a periodic orbit loses its stability as one of its eigenvalues (or a pair of eigenvalues) crosses out through the unit circle, and the global bifurcations (homoclinic bifurcations) where a connection is established from an unstable solution and back to the same solution along one of its stable directions. The band merging bifurcations considered in Sec. 1.3 for the logistic map and for the skew tent map are examples of homoclinic bifurcations.
The second type of bifurcations, referred to as border-collision bifurcations, are connected with situations where the trajectory starts to intersect one of the so-called sewing surfaces, i.e., surfaces that divide the phase space into domains of different dynamics. Within each such domain the system is smooth, but the equations of motion change abruptly from one domain to the next. This type of bifurcation, which typically involves abrupt jumps in the eigenvalues of the orbit, cannot occur in smooth dynamical systems.
A simple type of border-collision bifurcation consists in the continuous transformation of a solution from one type into another. However, more complicated phenomena are also possible, including new types of direct transition to chaos, and the merging...
