In this chapter the authors talk about importance of chaos synchronization on technology and science. This chapter is developed in three sections. In the former the authors tackle the subject related with the so-called the synchronized state in the sense of identical synchronization. It is done via robust nonlinear observer design, considering corrupted measurements and model uncertainties, coupling uncertainty estimators with nonlinear state observers. The second part treats the subject related to the applications to chaos communications, that is to say, an application of chaos theory which is aimed to provide security in the transmission of information performed through telecommunications technologies, speaking roughly at the transmitter, a message is added on to a chaotic signal and then, the message is masked in the chaotic signal. As it carries the information, the chaotic signal is also called chaotic carrier. This is done via control theory and is a particular case of chaos synchronization. Finally, in the latter section the authors talk about application to synchronization of biological systems that is to say, the intercellular Ca²+ waves have been seen like a mechanism by means of which a group of cells can communicate with one another and coordinate a multicellular response to a local event. Recently, it has been observed in a variety of systems that calcium signals can also propagate from one cell to another and thereby serve as a means of intercellular communication. The desire to understand the biophysical mechanisms of cellular dynamics has lead to introduce feedback control laws in some biological systems. Departing from the above ideas, this section explores links between feedback control schemes, with an external input, and intracellular calcium functions for coordination and control.
TopIdentical Master-Slave Synchronization
As is well known the study of the synchronization problem for chaotic oscillators has been very important from the nonlinear science point of view, in particular the applications to biology, medicine, cryptography, secure data transmission and so on.
In general, the synchronization research has been focused onto two areas, the first one, related with the employ of state observers, where the main applications lies on the synchronization of nonlinear oscillators with the same model structure and order, but different initial conditions and/or parameters (Celikovsky, 2005; Ling, 2004; Martinez-Guerra, 2008; Morgül, 1996; Nijmeijer, 1997a; Shihua, 2004; Solak, 2004).
As is well known observer schemes are widely used for the reconstruction of no measured state dynamics. The only available information is the measured system’s output, which represents a function of some current inner states of the system. Usually, the dimension of the vector of output measured signals is smaller than the dimension of the corresponding vector of states; therefore it is necessary to develop estimation techniques, known as observer design dealing with on-line state estimation.
The most successful observation schemes need a nominal model for their implementation, but as is well known the exact knowledge of the nonlinearities of nonlinear plants is a hard task, such that uncertain systems must be tackled. This situation leads to the standard observers not be realizable. Interesting research of observer based chaotic system synchronization have been done, (Mörgul, 1996), applied an observer for nonlinear oscillators which can be transformed in an observability canonical form, reduced order observers have been employed for synchronization purposes and parameters identification in chaotic oscillators (Martinez-Guerra, 2006; Mendoza-Camargo 2004; Aguilar-Ibañez, 2006) with adequate performance, recently in (Martinez-Guerra, 2006) a reduced order observer for observable uncertain chaotic oscillators via algebraic differential approach have been presented. (Alvarez, 2003) presented an observer design which estimate unobservable states of nonlinear plants under the assumption of nominal plant model knowledge, (Aguilar-Lopez, 2002; Aguilar, 2003; Martinez-Guerra, 2004) developed high gain observers for uncertainty estimation in nonlinear systems and integral-type observer for state estimation for partially unknown nonlinear plants, however the problem of state estimation with unobservable uncertainties still remains. Following these ideas, researches were oriented to the observation/estimation problem subjected to bounded nonlinearities or uncertainties. If the plant model is uncertain or incomplete, which is the most common case, the implementation of high gain observers turns out to be adequate. Besides, the designs of new robust observers based on adaptive techniques, such as neural networks, have been proposed (Huaguang, 2007; Pogromsky, 1998; Shihua, 2004).