Etude mathématique et numérique d’une équation différentielle d’ordre fractionnaire

Abstract

This thesis work is devoted to the study of fractional differential equations. Particular attention is being given to the fractional Selkov model, which is a non-linear system composed of fractional differential equations that allows the modeling of a phenomenon called "glycolysis". A detailed study of this system was carried out : the existence and the uniqueness of the solution, stability, discretization, bifurcation . . ., these last were evaluated in a numerical form through the phase portraits, the bifurcations diagram, and the Lyapunov exponent. On the other hand, in this thesis we are also interested in the problem of controlling and synchronizing chaos in utonomous and non-autonomous fractional differential equations. Based on some concepts of fractional alculus theory we present some techniques. Among the many strategies to achieve chaos synchronization in fractional chaotic systems we present : the modified projective adaptive synchronization technique. This method has the advantage of being applied to chaotic, hyperchaotic, identical, different, and uncertain systems and also chieves complete synchronization, anti-synchronization and projective synchronization. Finally, attention should be drawn to another equally interesting type of synchronization, the combination synchronization of non-utonomous fractional chaotic systems. Numerical results are given to verify the feasibility and effectiveness of the proposed combination synchronization method. This type of synchronization has the possibility of giving more security in secure communications. These results extend without difficulty if we apply this method to ractional systems of the same dimension