**7. Conclusion**

In this chapter, the case of unidirectional coupling scheme of two chaotic non-autonomous dynamical systems was studied. The proposed system is the second order Birkhoff-Shaw system, which is simple but very interesting from the perspective of nonlinear analysis. Furthermore, the coupling method was based on a recently new proposed scheme based on the nonlinear controller, which is applied for the first time in non-autonomous systems.

The Birkhoff-Shaw system is one of the simplest 2-D nonlinear systems exhibiting a rich dynamical behaviour. Besides limit cycles, Birkhoff-Shaw system presents quasiperiodicity and chaos, which can make the control of the system a difficult target in practical applications, where a particular dynamic is desired. Also, two well-known phenomena of nonlinear theory, the Intermittency and the Interior Crisis have been observed. However, the main drawback of this system is the fact that this system is a non-autonomous dynamical system, which makes the coupling method weak, especially if it is used in secure communication schemes.

In agreement to the simulation results, the circuital implementation of the proposed system in SPICE, in the case of unidirectional coupling, confirms the appearance of complete synchronization and antisynchronization, depending on the signs of the parameters of the error functions, in various cases. With this method, by choosing an appropriate sign for the error functions, the coupling system can be driven either to complete synchronization or antisynchronization behaviour.

From our knowledge, the complex behaviour of chaotic systems, like the ones that mentioned above, makes the synchronization difficult in practical applications where a particular dynamic is desired. For this reason, the synchronization of chaotic systems has attracted considerable attention due to its great potential applications, in secure communication, chemical reactions and biological systems. Especially, the synchronization in coupled neurons is a subject of a growing interest in the research community. So, due to the fact that Birkhoff-Shaw chaotic attractor exhibits the structure of beaks and wings, typically observed in chaotic neuronal models, the proposed coupling scheme showed an interesting research result of achieving the synchronization or antisynchronization in the case of coupled neuronal models.

As a next step in this direction is the application of the proposed method in non-identical Birkhoff-Shaw coupled systems in order to satisfy the goal of control of systems, which are in totally different dynamical behaviours. Also, the case of bidirectional coupling as well as the case of generalized synchronization, with the proposed scheme, could be examined.
