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The adaptive dual synchronization of chaotic (hyperchaotic) complex
systems with uncertain parameters has been investigated. The analytical
control functions are derived using a theorem to synchronize the chaotic
(hyperchaotic) solutions of these systems. The adaptive dual synchronization
between the chaotic complex Chen and Lorenz systems is introduced
as an example, and another example is used to test the validity
of the technique of this paper. Other examples of chaotic or hyperchaotic
complex systems can be similarly studied. Based on the up-to-date
laws, the parameters of the drive systems can be identified. The image encryption
technique based on the adaptive dual synchronization of chaotic
complex Chen and Lorenz systems is presented for gray and color images
in the same time. Meantime, in the receiver side, information can be recovered
successfully by adaptive technique. The presented technique is robust
with respect to different levels of white Gaussian noise. The communication
channel as well as the effect of the increase of noise are big challenge
which have not been considered. Numerical simulations are given to verify
the feasibility of our proposed synchronization and better performance of
image encryption technique in terms of histogram, robustness to noise and
visual imperceptibility.

The adaptive dual synchronization of chaotic (hyperchaotic) complex
systems with uncertain parameters has been investigated. The analytical
control functions are derived using a theorem to synchronize the chaotic
(hyperchaotic) solutions of these systems. The adaptive dual synchronization
between the chaotic complex Chen and Lorenz systems is introduced
as an example, and another example is used to test the validity
of the technique of this paper. Other examples of chaotic or hyperchaotic
complex systems can be similarly studied. Based on the up-to-date
laws, the parameters of the drive systems can be identified. The image encryption
technique based on the adaptive dual synchronization of chaotic
complex Chen and Lorenz systems is presented for gray and color images
in the same time. Meantime, in the receiver side, information can be recovered
successfully by adaptive technique. The presented technique is robust
with respect to different levels of white Gaussian noise. The communication
channel as well as the effect of the increase of noise are big challenge
which have not been considered. Numerical simulations are given to verify
the feasibility of our proposed synchronization and better performance of
image encryption technique in terms of histogram, robustness to noise and
visual imperceptibility.

The adaptive dual synchronization of chaotic (hyperchaotic) complex
systems with uncertain parameters has been investigated. The analytical
control functions are derived using a theorem to synchronize the chaotic
(hyperchaotic) solutions of these systems. The adaptive dual synchronization
between the chaotic complex Chen and Lorenz systems is introduced
as an example, and another example is used to test the validity
of the technique of this paper. Other examples of chaotic or hyperchaotic
complex systems can be similarly studied. Based on the up-to-date
laws, the parameters of the drive systems can be identified. The image encryption
technique based on the adaptive dual synchronization of chaotic
complex Chen and Lorenz systems is presented for gray and color images
in the same time. Meantime, in the receiver side, information can be recovered
successfully by adaptive technique. The presented technique is robust
with respect to different levels of white Gaussian noise. The communication
channel as well as the effect of the increase of noise are big challenge
which have not been considered. Numerical simulations are given to verify
the feasibility of our proposed synchronization and better performance of
image encryption technique in terms of histogram, robustness to noise and
visual imperceptibility.

The adaptive dual synchronization of chaotic (hyperchaotic) complex
systems with uncertain parameters has been investigated. The analytical
control functions are derived using a theorem to synchronize the chaotic
(hyperchaotic) solutions of these systems. The adaptive dual synchronization
between the chaotic complex Chen and Lorenz systems is introduced
as an example, and another example is used to test the validity
of the technique of this paper. Other examples of chaotic or hyperchaotic
complex systems can be similarly studied. Based on the up-to-date
laws, the parameters of the drive systems can be identified. The image encryption
technique based on the adaptive dual synchronization of chaotic
complex Chen and Lorenz systems is presented for gray and color images
in the same time. Meantime, in the receiver side, information can be recovered
successfully by adaptive technique. The presented technique is robust
with respect to different levels of white Gaussian noise. The communication
channel as well as the effect of the increase of noise are big challenge
which have not been considered. Numerical simulations are given to verify
the feasibility of our proposed synchronization and better performance of
image encryption technique in terms of histogram, robustness to noise and
visual imperceptibility.

A new chaotic system with line equilibrium is introduced in this paper. This system consists of five terms with two transcendental nonlinearities and two quadratic nonlinearities. Various tools of dynamical system such as phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagram and Poincarè map are used. It is interesting that this system has a line of fixed points and can display chaotic attractors. Next, this paper discusses control using passive control method. One example is given to insure the theoretical analysis. Finally, for the new chaotic system, an electronic circuit for realizing the chaotic system has been implemented. The numerical simulation by using MATLAB 2010 and implementation of circuit simulations by using MultiSIM 10.0 have been performed in this study.

In this article, we present a generalization of stability theorems for Caputo fractional derivative to the distributed fractional-order (DFO) case by using the Laplace transform and the asymptotical expansion of the generalized Mittag-Leffler function. We propose the definition of the generalized Wright stability to study the stability of DFO nonlinear dynamical system using Lyapunov direct method. The linear feedback control is used to stabilize a class of chaotic DFO nonlinear dynamical systems. Using Lyapunov direct method, we study the synchronization between two identical chaotic systems and between two other different in the linear terms. The chaotic DFO Lorenz system is given as an example to achieve the linear feedback control technique. Another two examples which are chaotic DFO complex Chen and L"{u} systems are used to show the validity and feasibility of our proposed synchronization scheme. Numerical simulations are implemented to verify the results of these investigations.

In this article, we present a generalization of stability theorems for Caputo fractional derivative to the distributed fractional-order (DFO) case by using the Laplace transform and the asymptotical expansion of the generalized Mittag-Leffler function. We propose the definition of the generalized Wright stability to study the stability of DFO nonlinear dynamical system using Lyapunov direct method. The linear feedback control is used to stabilize a class of chaotic DFO nonlinear dynamical systems. Using Lyapunov direct method, we study the synchronization between two identical chaotic systems and between two other different in the linear terms. The chaotic DFO Lorenz system is given as an example to achieve the linear feedback control technique. Another two examples which are chaotic DFO complex Chen and L"{u} systems are used to show the validity and feasibility of our proposed synchronization scheme. Numerical simulations are implemented to verify the results of these investigations.

In this article, we present a generalization of stability theorems for Caputo fractional derivative to the distributed fractional-order (DFO) case by using the Laplace transform and the asymptotical expansion of the generalized Mittag-Leffler function. We propose the definition of the generalized Wright stability to study the stability of DFO nonlinear dynamical system using Lyapunov direct method. The linear feedback control is used to stabilize a class of chaotic DFO nonlinear dynamical systems. Using Lyapunov direct method, we study the synchronization between two identical chaotic systems and between two other different in the linear terms. The chaotic DFO Lorenz system is given as an example to achieve the linear feedback control technique. Another two examples which are chaotic DFO complex Chen and L"{u} systems are used to show the validity and feasibility of our proposed synchronization scheme. Numerical simulations are implemented to verify the results of these investigations.

In this article, we present a generalization of stability theorems for Caputo fractional derivative to the distributed fractional-order (DFO) case by using the Laplace transform and the asymptotical expansion of the generalized Mittag-Leffler function. We propose the definition of the generalized Wright stability to study the stability of DFO nonlinear dynamical system using Lyapunov direct method. The linear feedback control is used to stabilize a class of chaotic DFO nonlinear dynamical systems. Using Lyapunov direct method, we study the synchronization between two identical chaotic systems and between two other different in the linear terms. The chaotic DFO Lorenz system is given as an example to achieve the linear feedback control technique. Another two examples which are chaotic DFO complex Chen and L"{u} systems are used to show the validity and feasibility of our proposed synchronization scheme. Numerical simulations are implemented to verify the results of these investigations.