Application to Chaos Control and Chaos Synchronization

Since Pecora and Carroll [117, 118] first designed an analog electrical circuit to realize chaos synchronization, many researchers have extensively studied the property of chaos synchronization and possible applications in practice. This has changed a long time viewpoint: chaos cannot be controlled, nor synchronized.

Although many results about chaos synchronization have been obtained on the basis of stability theory, general mathematical theory and methodology are still under development. Recently, Curran and Chua [20] suggested that different chaos synchronization methods should be unified to establish a fundamental mathematical theory on the basis of the absolute stability theory of Lurie control systems. The authors and their coworkers have also studied chaos synchronization following Curran and Chua’s idea [81, 82, 154]. For the Chua’s chaotic circuit, ywe have recently found that it could be transformed into a type of Lurie system, and thus the theory and methodology developed by Liao [72–77] can be used to study the synchronization of two Chua’s circuits. Chua’s circuit is the first electrical circuit to realize chaos in experiment, which exhibits very rich complex dynamical behavior, and yet has very high potential in real applications.

In this chapter, as an application, we will apply the absolute stability of Lurie control systems developed in previous chapters to study the globally exponential synchronization of two Chua’s chaotic circuits [95]. Also we propose and develop the theory and methodology of absolutely exponential stability, and investigate the global synchronization of two chaotic systems with feedback controls [81, 82, 89]. The materials presented in this chapter are mainly chosen from Liao and Yu [95] (Sects. 13.1–13.5), Liao et al. [89] (Sect. 13.6.1), and Liao et al. [84] (Sect. 13.6.2).