TABLE OF CONTENTS The purpose of this chapter is to discuss a number of new phenomena that can arise in piecewise-smooth dynamical systems whose motion involves two (or more) periodic components. When two self-oscillatory systems interact (or a self-oscillatory system is subjected to an external periodic forcing), the total motion can be viewed as occurring on the surface of a two-dimensional torus and, in the absence of resonances, the motion is said to be quasiperiodic. The problem that we are interested in is associated with the role that border-collision bifurcations play in the synchronization of the two modes and in the transition to chaos when the torus finally breaks down.
To examine these phenomena we shall consider a three-dimensional model of a DC/DC converter with pulse-width modulation. As described in Chapter 5, such converters are widely used in power electronics where they are applied in power supplies for industrial and aerospace equipment as well as in domestic appliances [1, 2, 3, 4, 5].
Synchronization is a universal phenomena in nonlinear systems [6]. The classical example is the synchronization of two mechanical clocks hanging side by side on the same wall. In his attempts to evaluate the accuracy of the newly developed pendulum clock, Huygens [7] observed that when two clocks could influence each others motion via imperceptible vibrations in the air or in the common support, the clocks tended to work in synchrony. Similar phenomena also occur in a wide range of different...
