Emad E. Mahmoud

ACTIVE CONTROL AND GLOBAL SYNCHRONIZATION OF THE COMPLEX CHEN AND LÜ SYSTEMS

Chaos synchronization is a very important nonlinear phenomenon, which has been studied to date extensively on dynamical systems described by real variables. There also exist, however, interesting cases of dynamical systems, where the main variables participating in the dynamics are complex, for example, when amplitudes of electromagnetic fields are involved. Another example is when chaos synchronization is used for communications, where doubling the number of variables may be used to increase the content and security of the transmitted information. It is also well-known that similar generalization of the Lorenz system to one with complex ODEs has been introduced to describe and simulate the physics of a detuned laser and thermal convection of liquid flows. In this paper, we study chaos synchronization by applying active control and Lyapunov function analysis to two such systems introduced by Chen and Lu. First we show that, written in terms of complex variables, these systems can have chaotic dynamics and...

ANALYSIS OF HYPERCHAOTIC COMPLEX LORENZ SYSTEMS

This paper introduces and analyzes new hyperchaotic complex Lorenz systems. These systems are 6-dimensional systems of real first order autonomous differential equations and their dynamics are very complicated and rich. In this study we extend the idea of adding state feedback control and introduce the complex periodic forces to generate hyperchaotic behaviors. The fractional Lyapunov dimension of the hyperchaotic attractors of these systems is calculated. Bifurcation analysis is used to demonstrate chaotic and hyperchaotic behaviors of our new systems. Dynamical systems where the main variables are complex appear in many important fields of physics and communications.

Synchronization and control of hyperchaotic complex Lorenz system

The aim of this paper is to investigate the phenomenon of projective synchronization (PS) and modified projective synchronization (MPS) of hyperchaotic attractors of hyperchaotic complex Lorenz system which has been introduced recently in our work. The control problem of these attractors is also studied. Our system is a 6-dimensional continuous real autonomous hyperchaotic system. The active control method based on Lyapunov function is used to study PS and MPS of this system. The problem of hyperchaos control is treated by adding the complex periodic forcing. The control performances are verified by calculating Lyapunov exponents. Numerical simulations are implemented to verify the results of these investigations.

Dynamics and synchronization of new hyperchaotic complex Lorenz system

Abstract The aim of this paper is to introduce a new hyperchaotic complex Lorenz system. This hyperchaotic complex system is constructed by adding a linear controller to the second equation of the chaotic complex Lorenz system. The new system is a 7-dimensional continuous real autonomous hyperchaotic system. This system has hyperchaotic attractors and quasi-periodic solutions with three zero Lyapunov exponents, while the chaotic attractors exist for all the parameters values of this system with two zero Lyapunov exponents. The fractional Lyapunov dimension of the hyperchaotic attractors of this system is calculated. Bifurcation diagrams are used to demonstrate chaotic and hyperchaotic behaviors of new system. The active control method based on Lyapunov stability analysis is used to study synchronization of this system. Numerical simulations are implemented to verify the results of these investigations.

Adaptive anti-lag synchronization of two identical or non-identical hyperchaotic complex nonlinear systems with uncertain parameters

In this paper we present the adaptive anti-lag synchronization (ALS) of two identical or non-identical hyperchaotic complex nonlinear systems with uncertain parameters. The concept of ALS is not detected yet in the literature. Based on the Lyapunov function a scheme is designed to achieve ALS of hyperchaotic attractors of these systems. The ALS of two identical complex Lu systems and two different hyperchaotic complex Lorenz and Lu systems are taken as two examples to verify the feasibility of the presented scheme. These hyperchaotic complex systems appear in several applications in physics, engineering and other applied sciences. Numerical simulations are calculated to demonstrate the effectiveness of the proposed synchronization scheme and verify the theoretical results.

On projective synchronization of hyperchaotic complex nonlinear systems based on passive theory for secure communications

In this paper we deal with the projective synchronization (PS) of hyperchaotic complex nonlinear systems and its application in secure communications based on passive theory. The unpredictability of the scaling factor in PS can additionally enhance the security of communications. In this paper, a scheme for secure message transmission is proposed, and we try to transmit more than one large or bounded message from the transmitter to the receiver. The new hyperchaotic complex Lorenz system is employed to encrypt these messages. In the transmitter, the original messages are modulated into its parameter. In the receiver, we assume that the parameter of the receiver system is uncertain. The controllers and corresponding parameter update law are constructed to achieve PS between the transmitter and receiver system with an uncertain parameter, and identify the unknown parameter via passive theory. The original messages can be recovered successfully through some simple operations by the estimated parameter. Numerical results have verified the effectiveness and feasibility of the presented method.

Modified projective phase synchronization of chaotic complex nonlinear systems

This paper introduces the concept of Modified Projective Phase Synchronization (MPPS) for interacting chaotic systems with complex variables. The idea is that the number of effective state variables can be increased by treating the real and imaginary parts separately. On the basis of the Lyapunov stability theory, a scheme is designed to realize the new form of chaotic synchronization, and we demonstrate how chaotic complex systems in a master–slave configuration can be synchronized to a constant scaling matrix. The speed and accuracy of the synchronization are illustrated by means of computer simulation.

MODIFIED PROJECTIVE LAG SYNCHRONIZATION OF TWO NONIDENTICAL HYPERCHAOTIC COMPLEX NONLINEAR SYSTEMS

In this work, we introduce and investigate the modified projective lag synchronization (MPLS) of two nonidentical hyperchaotic complex nonlinear systems. The idea of an active control technique based on complex Lyapunov function with lag in time is used for an approach to investigate MPLS of hyperchaotic attractors of these systems. For illustration, this approach is applied to hyperchaotic complex Chen and Lu systems. Numerical results are calculated to test the validity of the analytical expressions of control functions to achieve MPLS.

Passive control of n-dimensional chaotic complex nonlinear systems

In this paper we investigate the control of n-dimensional chaotic complex nonlinear systems by using passive control theory. These complex systems have been introduced and studied recently in our investigations. Based on the property of the passive system, an approach is stated to design the passive controller and realize the control of these systems. As an example, we apply this approach to convert the chaotic attractors of the complex Lü system into the trivial, nontrivial equilibrium points, periodic (limit cycle) and quasi-periodic solutions. This example appears in several fields of physics and engineering, e.g. nonlinear electronic circuits and communications. A block diagram of this example using Matlab/Simulink is constructed. The analytical results of the controllers which have been calculated by using this approach are tested numerically and good agreement is obtained.

Controlling hyperchaotic complex systems with unknown parameters based on adaptive passive method

The aim of this paper is to study the control of hyperchaotic complex nonlinear systems with unknown parameters using passive control theory. An approach is stated to design the passive controller and estimate the unknown parameters based on the property of the passive system. The feasibility and effectiveness of the proposed approach is demonstrated through its application to the hyperchaotic complex Lu system, as an example. The estimated values of the unknown parameters are calculated. The analytical form of the complex controller is derived and used in the numerical simulation to control the hyperchaotic attractors of this example. Block diagrams of this example using Matlab/Simulink are constructed after and before the control to ensure the validity of the analytical results. Other examples of hyperchaotic complex nonlinear systems can be similarly treated.