الفهرس 
Acknowledgments
The variational iteration methodOnly 14 pages are availabe for public view
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Abstract
The main goal of this thesis is to study chaos synchronization in some fractional-order chaotic and hyperchaotic systems. Based on the stability theory of fractional-order differential systems, we have derived some Routh-Hurwitz conditions to study local stability in fractional-order systems. These fractional Routh-Hurwitz criteria had been utilized to synchronize only fractional-order systems. These results are original and have not been published by others before.
This thesis consists of five chapters:
Chapter one is devoted to introduce the preliminaries. We have introduced a review of some basic concepts about the integer and fractional-order dynamical systems such as stability, periodic solutions, bifurcations, attractor, strange attractor, chaos, hyperchaos and Lyapunov exponents. In chapter two, we have investigated stability, bifurcations, chaos and synchronization of the Liu system and its fractional-order counterpart. We have achieved chaos synchronization of the fractional-order Liu system via unidirectional linear error feedback coupling method and the function projective synchronization between the commensurate fractional-order Liu system and its integer order form have been obtained. Furthermore, the effect of fractional-order on the synchronization of the commensurate fractional-order Liu system has been shown. In chapter three, we have analyzed some dynamical behaviors of the fractional-order financial system. In addition, we have studied the local stability of the equilibria using the fractional Routh-Hurwitz conditions. Furthermore, chaos has been shown to exist in this system with orders less than three. Analytical conditions for linear feedback control have been achieved, showing the effect of the fractional-order on controlling chaos in this system. The effect of fractional-order on chaos synchronization has been demonstrated as well by using nonlinear feedback controllers. In chapter four, we have derived some Routh-Hurwitz criteria for the stability in fractional-order hyperchaotic systems. These conditions have been applied to the novel fractional-order hyperchaotic system and the fractional-order hyperchaotic Chen system. Numerical simulations and Lyapunov exponents have been used to show that hyperchaos exists in these systems with order less than 4. The theoretical conditions for controlling hyperchaos in the novel fractional-order hyperchaotic system have been obtained using linear feedback control technique. The efficiency of applying the proposed technique to control only fractional-order hyperchaotic systems has been shown. It has also been shown that the novel fractional-order hyperchaotic system has been synchronized using feedback control method but its integer-order counterpart has not been synchronized using the same controllers. Based on the Laplace transformation theory, we have achieved synchronization of the fractional-order hyperchaotic Chen system when choosing suitable linear controllers. Finally, the fractional-order hyperchaotic Chen system has been synchronized using nonlinear control method but its integer-order counterpart has not been synchronized using the same nonlinear controllers. In chapter five, chaos synchronization between two different chaotic (hyperchaotic) systems has been demonstrated. Chaos synchronization between two different fractional-order chaotic systems has been studied using linear control technique; the fractional-order Chen system has been used to drive fractional-order Lü system and the fractional-order Lorenz-like system has been used to drive the fractional-order Chen system. Conditions for chaos synchronization have been investigated theoretically by using the Laplace transform. Moreover, the novel fractional-order hyperchaotic system has been used to drive the fractional-order hyperchaotic Chen system.
Throughout this thesis, the simulations results are carried out using an efficient method for solving fractional-order differential equations that is; the predictor-correctors scheme which represents a generalization of the Adams-Bashforth-Moulton algorithm. In chapters 2, 3, 4 and 5 we have used the numerical simulations to show the effectiveness of the proposed synchronization techniques.
We summarize the new results given in this thesis as follow:
(1) We have derived some Routh-Hurwitz conditions for the local stability of fractional-order hyperchaotic systems [63,65]; (2) We have proposed the novel fractional-order hyperchaotic system [63,126], and the fractional-order hyperchaotic Chen system [65,126], then we studied their synchronization [63,64,65,126]; (3) We have applied the fractional Routh-Hurwitz conditions to study the effect of fractional-order on synchronization of the fractional-order Liu chaotic system [115,116,118], the novel and Chen fractional-order hyperchaotic systems [63,64,65,126]. We have shown the effect of fractional-order on chaos control of the fractional-order financial system [120] as well; (4) chaos synchronization between two different chaotic (hyperchaotic) systems
has been demonstrated [126].