The Method of Schwarzian Derivatives

The essential operation that characterizes most power electronics circuits is the cyclic switching of the circuit con guration from one linear system to another. Such periodic switching operation naturally permits discrete-time modeling of power electronics systems in the form of iterative functions. In this section we review iterative functions of rst order and in particular discuss a technique for deciding if a given iterative function can period-double to chaos. The main tool that we use for this study is the Schwarzian derivative, which was de ned by the German mathematician Hermann Schwarz in 1869 for studying complex value functions. Moreover, our purpose here is to try to make use of certain important properties of the Schwarzian derivative for constructing a necessary condition for the occurrence of period-doubling cascades in rst-order iterative functions. As we will see, this technique nds application in establishing the possibility of a route to chaos via period-doubling in dc/dc converters that operate in discontinuous mode.