Systèmes dynamiques et chaos

Abstract

In this thesis we have addressed the problem of the control and the synchronization of the dynamic chaotic systems. This thesis is broken up into four parts. In the first part of the thesis we have given the definitions and characteristics of the dynamic and chaotic systems, leading to a definition that is going to allow to the scientists an understanding and an application more increased of the chaotic systems, one finds fundamental theoretical concepts for the analysis of the behavior and different classic methods of control of these systems. In the second part, we discussed chaos control problems, the mathematical formulation of the control of chaotic processes. We have also proposed in this part the most reliable and well-known methods of controlling chaotic processes : the OGY method, the TDFC method, and the non-linear adaptive control method, these methods aim to improve the suppression of chaos and drive the system to converge to the desired periodic orbit or stabilize at its fixed point. In the last chapter of this thesis the problem of synchronization of chaotic systems was addressed, we have presented several synchronization schemes, two techniques have been presented to achieve identical synchronization where two coupled chaotic systems exhibit identical oscillations : Pecora and Carroll technique, and the second technique is based on a linear coupling. Subsequently, a generalized synchronization method was presented. This approach is an extension of the previous phenomenon involving the presence of functional relationships between the two coupled systems. Finally, An adaptive control scheme is designed for the modified projective synchronization (AMPS) of two chaotic systems based on Lyapunov stability theorem, in this method the drive and response systems could be synchronized up to a constant scaling matrix, a numerical examples are given to demonstrate the effectiveness of the proposed method. This latter method is important because it includes complete synchronization, antisynchronization and projective synchronization. The advantage of this method is that it is applied for the synchronization of identical systems, different systems, hyperchaotic systems and uncertain systems.