**1. Introduction**

In the past decades, the phenomenon of synchronization between coupled nonlinear systems and especially of systems with chaotic behaviour has attracted scientists' interest from all over

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

the world because it is an interesting phenomenon with a broad range of applications, such as in various complex physical, chemical and biological systems [1–9], in secure and broadband communication system [10, 11] and in cryptography [12, 13].

In synchronization two or more systems with chaotic behaviour can adjust a given of their motion property to a common behaviour (equal trajectories or phase locking), due to forcing or coupling [14]. However, having two chaotic systems being synchronized, it is a major surprise, due to the exponential divergence of the nearby trajectories of the systems. Nevertheless, nowadays the phenomenon of synchronization of coupled chaotic oscillators is wellstudied theoretically and proven experimentally.

Synchronization theory has begun studying in the 1980s and early 1990s by Fujisaka and Yamada [15], Pikovsky [16], Pecora and Carroll [17]. Onwards, a great number of research works based on synchronization of nonlinear systems has risen and many synchronization schemes depending on the nature of the coupling schemes and of the interacting systems have been presented. Complete or full chaotic synchronization [18–23], phase synchronization [24, 25], lag synchronization [26, 27], generalized synchronization [28], antisynchronization [29, 30], anti-phase synchronization [31–36], projective synchronization [37], anticipating [38] and inverse lag synchronization [39] are the most interesting types of synchronization, which have been investigated numerically and experimentally by many research groups.

This chapter deals with two of the aforementioned cases: the complete synchronization and the antisynchronization. In the case of complete synchronization, two identically coupled chaotic systems have a perfect coincidence of their chaotic trajectories, i.e., *x*1(*t*) = *x*2(*t*) as *t* → ∞. In the case of antisynchronization, for initial conditions chosen from large regions in the phase space two coupled systems *x*1 and *x*2, can be synchronized in amplitude, but with opposite sign, that is *x*1(*t*) = –*x*2(*t*) as *t* → ∞.

From our knowledge, chaotic systems exhibit high sensitivity on initial conditions or system's parameters and if they are identical and start from almost the same initial conditions, they follow trajectories which rapidly become uncorrelated. That is why many techniques exist to obtain chaotic synchronization. So, many of these techniques for coupling two or more nonlinear chaotic systems can be mainly divided into two classes: unidirectional coupling and bidirectional or mutual coupling [40]. In the first case, only the first system, the master system, drives the second one, the slave system, while in the second case, each system's dynamic behaviour influences the dynamics of the other.

Furthermore, the subject of synchronization between coupled chaotic systems, especially in the last decade, plays a crucial role in the field of neuronal dynamics [6, 41]. Neural signals in the brain are observed to be chaotic and it is worth considering further their possible synchronization [42–46]. These signals are produced by nerve membranes exhibiting their own nonlinear dynamics, which generate and propagate action potentials. Such nonlinear dynamics in nerve membranes can produce chaos in neurons and related bifurcations.

So, motivated by the aforementioned fact, the Birkhoff-Shaw system [45], which exhibits the structure of beaks and wings, typically observed in chaotic neuronal models, is chosen for use in this chapter. It is a second order non-autonomous dynamical system with rich dynamical behaviour, which has not been sufficiently studied. Furthermore, the unidirectional coupling scheme, which is used, is designed by using the nonlinear controllers to target synchronization states, such as complete synchronization and antisynchronization, with amplification or attenuation in chaotic oscillators. The stability of synchronization is ensured by using Lyapunov function stability theorem in the unidirectional mode of coupling. The simulation results from system's numerical integration confirm the appearance of complete synchronization and antisynchronization phenomena depending on the signs of the parameters of the error functions. Electronic circuitry that models the coupling scheme is also reported to verify its feasibility.

This chapter is organized as follows. In Section 2, the features of chaotic systems and especially of the proposed Birkhoff-Shaw system by using various tools of nonlinear theory, such as bifurcation diagrams, phase portraits and Lyapunov exponents, are explored. The synchronization scheme, by using the nonlinear controller, as well as the unidirectional coupling scheme is discussed in Sections 3 and 4, respectively. The simulation results of the proposed method are presented for various cases in Section 5. Section 6 presents the circuital implementation of the coupling scheme and the results which are obtained by using the SPICE. Finally, the conclusive remarks and some thoughts for future works are drawn in the last section.
