In the preceding chapters we have been mainly concerned with studying the properties of chaotic dynamical systems. In this chapter we adopt a different, more active, point of view. In particular, we ask, can we use our knowledge of chaotic systems to achieve some desired goal? Two general areas where this point of view has proven useful are the control of chaos and the synchronization of chaotic systems. By control we shall generally mean feedback control. That is, we have some control variable that we can vary as a function of time, and we decide how to do this variation on the basis of knowledge (perhaps limited) of the system's past history and/or current state. In the synchronization of chaos, we generally shall be considering two (or more) systems that are coupled. The evolution is chaotic, and we are interested in the conditions such that the component systems execute the same motion. Both control of chaos and synchronization of chaos have potential practical applications, and we shall indicate these as this chapter proceeds.

Control of chaos

Two complementary attributes sometimes used to define chaos are (i) exponentially sensitive dependence and (ii) complex orbit structure. A quantifier for attribute (i) is the largest Lyapunov exponent, while a quantifier for attribute (ii) is the entropy (e.g., the metric entropy or the topological entropy, Section 4.5). These attributes of chaos can be exploited to fashion chaos-control strategies.