Piecewise-smooth Maps

Abstract

This course is about piecewise-smooth maps. If the phase space (typically \( \mathbb{R}^{n}\)) is partitioned into N disjoint open regions such that the union of the closures of these regions is the whole space, then a piecewise-smooth map is a map on this partition which is defined by a different smooth function on each region. Note that a piecewise-smooth map may be discontinuous across boundaries, or it may be continuous but the Jacobian matrix is discontinuous. Other classes exist, but these two form the basis for most studies. The decision about how to define dynamics on the boundaries of the regions can be a bit awkward and will involve us in some little technical issues later.