The goal of this chapter is to describe the basic properties of the complete synchronization of chaotic systems. Our approach is the following: we take a system as simple as possible and describe it in as detailed a way as possible. The simplest chaotic system is a one-dimensional map, it will be our example here. We start with the construction of the coupled map model, and describe phenomenologically, what complete synchronization looks like. The most interesting and nontrivial phenomenon here is the synchronization transition. We will treat it as a transition inside chaos, and follow a twofold approach. On one hand, we exploit the irregularity of chaos and describe the transition statistically. On the other hand, we explore deterministic regular properties of the dynamics and describe the transition topologically, as a bifurcation. We hope to convince the reader that these two approaches complement each other, giving the full picture of the phenomenon. In the next chapter, where we discuss many generalizations of the simplest model, we will see that the basic features of complete synchronization are generally valid for a broad class of chaotic systems.

The prerequisite for this chapter is basic knowledge of the theory of chaos, in particular of Lyapunov exponents. For the analytical statistical description we use the thermodynamic formalism, while for the topological considerations a knowledge of bifurcation theory is helpful. One can find these topics in many textbooks on nonlinear dynamics and chaos [Schuster 1988; Ott 1992; Kaplan and Glass 1995; Alligood etal. 1997; Guckenheimer and Holmes 1986] as well as in monographs [Badii and Politi 1997; Beck and Schlögl 1997].