Shaban Aly

Basic Properties And Chaotic Synchronization Of Complex Lorenz System

This paper aims at studying the basic properties and chaotic synchronization of complex Lorenz system: where α, γ, β are positive (real or complex) parameters, x and y are complex variables, z is a real variable, an overbar denotes complex conjugate variable and dots represent derivatives with respect to time. This system arises in many important applications in physics, for example, in laser physics and rotating fluids dynamics. Numerically we show that this system is a chaotic system and exhibits chaotic attractors. The necessary conditions for system (⋆) to generate chaos are obtained. Analytical and numerical calculations are presented to achieve synchronization. Active control technique is used to synchronize chaotic attractors of equations (⋆).

Strange attractors and chaos control in periodically forced complex Duffing's oscillators

An interesting and challenging research subject in the field of nonlinear dynamics is the study of chaotic behavior in systems of more than two degrees of freedom. In this work we study fixed points, strange attractors, chaotic behavior and the problem of chaos control for complex Duffings oscillators which represent periodically forced systems of two degrees of freedom. We produce plots of Poincare map and study the fixed points and strange attractors of our oscillators. The presence of chaotic behavior in these models is verified by the existence of positive maximal Lyapunov exponent. We also calculate the power spectrum and consider its implications regarding the properties of the dynamics. The problem of controlling chaos for these oscillators is studied using a method introduced by Pyragas (Phys. Lett. A 170 (1992) 421), which is based on the construction of a special form of a time-continuous perturbation. The study of coupled periodically forced oscillators is of interest to several fields of physics, mechanics and engineering. The connection of our oscillators to the nonlinear Schrodinger equation is discussed.

On periodic solutions of parametrically excited complex non-linear dynamical systems

An approximate analytical method, based on the generalized averaging method is extended to study periodic solutions of parametrically excited complex non-linear dynamical systems. It is well known that a great many problems of applied sciences often lead to the study of these dynamical systems. Our analytical approach provides us with specific values for the parameters of these dynamical systems for which such periodic solutions exist. An example which is related to rotor dynamics and spherical pendulum with vertically oscillating support is considered to illustrate this approach. Analytical results on this example are compared with numerical ones and excellent agreement is found between them.

Periodic attractors of complex damped non-linear systems

Abstract The aim of the present paper is to investigate the dynamics of a class of complex damped non-linear systems described by the equation (∗) z +ω 2 z+e z f(z, z , z , z )P( Ω t)=0, where z(t)=x(t)+ i y(t), i = −1 , the bar denotes the complex conjugate and e is a small positive parameter. The periodic attractors of Eq. (∗) are important in the study of these systems, since they represent stationary or repeatable behavior. This equation appears in several fields of physics, mechanics and engineering, for example, in high-energy accelerators, rotor dynamics, robots and shells. In the numerical investigation of this work we used the indicatrix method which has been introduced and extended in our previous studies to study the existence of the periodic attractors of our systems. To illustrate these periodic attractors we constructed Poincare plots at some of the parameter values which are obtained by the indicatrix method for the case ω 2 ≅ 1 4 , f=|z| 2 and P( Ω t)= sin 2t as an example. Our recent method which is based on the generalized averaging method is used to obtain approximate analytical solutions of Eq. (∗) and investigate the stability properties of the solutions. We compared the analytical results of our example with the numerical results and excellent agreement is found.

Dynamical properties and synchronization of complex non-linear equations for detuned lasers

We study the dynamics and synchronization properties of a system of complex non-linear equations describing detuned lasers. These equations possess a whole circle of fixed points, while the corresponding real variable equations have only isolated fixed points. We examine the stability of their equilibrium points and determine conditions under which the complex equations have positive, negative or zero Lyapunov exponents and chaotic, quasiperiodic or periodic attractors for a wide range of parameter values. We investigate the synchronization of chaotic solutions of our detuned laser system, using as a drive a similar set of equations and applying the method of global synchronization. We find attractors whose three-dimensional projection is not at all similar to the well-known shape of the (real) Lorenz attractor. Finally, we apply complex periodic driving to the electric field equation and show that the model can exhibit a transition from chaotic to quasiperiodic oscillations. This leads us to the discovery of an exact periodic solution, whose amplitude and frequency depend on the parameters of the system. Since this solution is stable for a wide range of parameter values, it may be used to control the system by entraining it with the applied periodic forcing.

Impulsive Control and Synchronization of Complex Lorenz Systems

In this paper, we continue our investigations on control and synchronization of the complex Lorenz systems by investigating impulsive control and synchronization. Nonlinear systems involving impulse effects, appear as a natural description of observed evolution phenomena of several real world problems; For example, many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, population dynamics and so forth do exhibit impulsive effects. Some new and more comprehensive criteria for global exponential stability and asymptotical stability of impulsively controlled complex Lorenz systems are established with varying impulsive intervals. The effectiveness of the proposed technique is verified through numerical simulations.

Turing instability in two-patch predator-prey population dynamics

In this paper, a spatio-temporal model as systems of ODE which describe two-species Beddington-DeAngelis type predatorprey system living in a habitat of two identical patches linked by migration is investigated. It is assumed in the model that the per capita migration rate of each species is influenced not only by its own but also by the other one’s density, i.e., there is cross diffusion present. We show that a standard (self-diffusion) system may be either stable or unstable, a cross-diffusion response can stabilize an unstable standard system and destabilize a stable standard system. For the diffusively stable model, numerical studies show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation and the cross migration response is an important factor that should not be ignored when pattern emerges.

Turing instability in a diffusive SIS epidemiological model

Modeling and simulation of infectious diseases help to predict the likely outcome of an epidemic. In this paper, a spatial susceptible-infective-susceptible (SIS) type of epidemiological disease model with self- and cross-diffusion are investigated. We study the effect of diffusion on the stability of the endemic equilibrium with disease-induced mortality and nonlinear incidence rate. In the absence of diffusion the stationary solution stays stable but becomes unstable with respect to diffusion and that Turing instability takes place. We show that a standard (self-diffusion) system may be either stable or unstable, cross-diffusion response can stabilize an unstable standard system or decrease a Turing space (the space which the emergence of spatial patterns is holding) compared to the Turing space with self-diffusion, i.e. the cross-diffusion response is an important factor that should not be ignored when pattern emerges. Numerical simulations are provided to illustrate and extend the theoretical results.