ABSTRACT
In this chapter, we discuss the different methods used for the rigorous proof of chaos in piecewise maps. The importance of these maps can be seen from the fact that the dynamics of a number of switching circuits can be represented by one-dimensional piecewise smooth maps under discrete modeling, in particular in power electronic circuits (Wolf, et al., (1994), Banerjee and Chakrabarty (1998), Banerjee, et al., (2000), Robert and Robert (2002)). The study of chaos in this type of system is based on the affinity of the corresponding normal forms for fixed points on borders, and the behavior of fixed points and periodic points depending on the bifurcation parameter for the scenarios associated with the various cases. A detailed overview of these maps is presented in (Zeraoulia and Sprott (2011(g))).
In Sec. 3.1, we present the essential results of the chaotic dynamics and bifurcations in 1-D piecewise smooth maps. In Sec. 3.2, the dynamical properties of an 1-D singular map are given and discussed along some rare types of bifurcations. Robust chaos and several types of border collision bifurcations are discussed in Sec. 3.3.
