Ahmed A. M. Farghaly

Chaotic and hyperchaotic attractors of a complex nonlinear system

In this paper, we introduce a complex nonlinear hyperchaotic system which is a five-dimensional system of nonlinear autonomous differential equations. This system exhibits both chaotic and hyperchaotic behavior and its dynamics is very rich. Based on the Lyapunov exponents, the parameter values at which this system has chaotic, hyperchaotic attractors, periodic and quasi-periodic solutions and solutions that approach fixed points are calculated. The stability analysis of these fixed points is carried out. The fractional Lyapunov dimension of both chaotic and hyperchaotic attractors is calculated. Some figures are presented to show our results. Hyperchaos synchronization is studied analytically as well as numerically, and excellent agreement is found.

CHAOTIC BEHAVIOR AND CHAOS CONTROL FOR A CLASS OF COMPLEX PARTIAL DIFFERENTIAL EQUATIONS

Systems of complex partial differential equations, which include the famous nonlinear Schrodinger, complex Ginzburg–Landau and Nagumo equations, as examples, are important from a practical point of view. These equations appear in many important fields of physics. The goal of this paper is to concentrate on this class of complex partial differential equations and study the fixed points and their stability analytically, the chaotic behavior and chaos control of their unstable periodic solutions. The presence of chaotic behavior in this class is verified by the existence of positive maximal Lyapunov exponent.The problem of chaos control is treated by applying the method of Pyragas. Some conditions on the parameters of the systems are obtained analytically under which the fixed points are stable (or unstable).

AN ACTIVE CONTROL FOR CHAOS SYNCHRONIZATION OF REAL AND COMPLEX VAN DER POL OSCILLATORS

In a recent paper [Chaos, Solitons Fractals21, 915 (2004)], both real and complex Van der Pol oscillators were introduced and shown to exhibit chaotic limit cycles. In the present work these oscillators are synchronized by applying an active control technique. Based on Lyapunov function, the control input vectors are chosen and activated to achieve synchronization. The feasibility and effectiveness of the proposed technique are verified through numerical simulations.

Stabilization of unstable periodic attractors of complex damped non-linear dynamical systems

Abstract In the recent paper [Int. J. Non-Linear Mech. 35 (2000) 309], periodic attractors were studied for a class of complex damped non-linear dynamical systems of the form: (*) z +ω 2 z+e z f(z, z , z , z )P(Ωt)=0, where z(t)=x(t)+iy(t), i = −1 , the bar denotes the complex conjugate and e is a small positive parameter. The object of this paper is to continue our investigation of this class by studying the stabilization of unstable periodic attractors of system (*) . A feedback control method which is suggested by Pyragas is used to stabilize these unstable periodic attractors. We construct Poincare plots before and after control to prove that the stabilization is achieved. Two examples are presented which served to illustrate this investigation. Maximal Lyapunov exponents and the power spectrum are calculated to show that our periodic attractors are not chaotic attractors.

A Technique for Studying a Class of Fractional-Order Nonlinear Dynamical Systems

In this work, we propose a technique to study nonlinear dynamical systems with fractional-order. The main idea of this technique is to transform the fractional-order dynamical system to the integer...

Double compound combination synchronization among eight n-dimensional chaotic systems

Depending on double compound synchronization and compound combination synchronization, a new kind of synchronization is introduced which is the double compound combination synchronization (DCCS) of eight n-dimensional chaotic systems. This kind may be considered as a generalization of many types of synchronization. In the communication, based on many of drive and response systems, the transmitted and received signals will be more secure. Using the Lyapunov stability theory and nonlinear feedback control, analytical formulas of control functions are obtained to insure our results. The corresponding analytical expression and numerical treatment are used to show the validity and feasibility of our proposed synchronization scheme. The eight memristor-based Chua oscillators are considered as an example. Other examples can be similarly investigated. The proposed synchronization technique is supported using the MATLAB simulation outcomes. We obtain the same results of numerical treatment of our synchronization using simulation observations of our example.

On Stabilization Of Solutions Of Complex Coupled Nonlinear Schrödinger Equations

We study the stabilization of nonchaotic periodic and quasi-periodic solutions of both integrable (α=1) and nonintegrable (α=2/3) of CCNLS equations of the form: \begin{eqnarray*} ip_t +p_{xx} +\frac{1}{2} \sigma (| p |^2 +\alpha | q |^2) p &= & \gamma g_1 (x)\exp (-i\omega_1t)\,,\\[5pt] iq_t +q_{xx} +\frac{1}{2} \sigma (\alpha | p |^2 +| q |^2) q &= & \gamma g_2 (x)\exp (-i\omega_2t)\,, \end{eqnarray*} where subscripts mean partial derivatives, p(x,t) and q(x,t) are the orthogonal components of an electric field in a glass fiber,

On chaos synchronization of a complex two coupled dynamos system

i=\sqrt{-1}

Chaos control of chaotic limit cycles of real and complex van der Pol oscillators

, the defocusing (σ=-1) and focusing (σ=1) cases are distinguished by σ; g1(x) and g2(x) are periodic functions in x and γ; and ω1 and ω2 are parameters. These solutions do not display sensitive dependence on initial conditions. The stabilization of solutions are studied using a feedback control method and their maximal Lyapunov exponents are calculated. Periodic solutions of this system are important in the study of these coupled equations, since they represent sta...