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 Lü. First we show that, written in terms of complex variables, these systems can have chaotic dynamics and exhibit strange attractors. We calculate numerically the values of the parameters at which these attractors exist. Active control and global synchronization techniques are then applied to study the phenomenon of chaos synchronization. Analytical criteria concerning the stability of these techniques are implemented and excellent agreement is found upon comparison with numerical experiments. In particular, studying the time evolution of "errors" (or differences between drive and control dynamics), we show that both techniques are very effective for controlling the behavior of these systems, even in regimes of very strong chaos.

Our main objective in this work is to investigate complete synchronization (CS) of n-dimensional chaotic complex systems with uncertain parameters. An adaptive control scheme is designed to study the synchronization of chaotic attractors of these systems. We applied this scheme, as an example, to study complete synchronization of chaotic attractors of two identical complex Lorenz systems. The adaptive control functions and the parameters estimation laws are calculated analytically based on the complex Lyapunov function. We show that the error dynamical systems are globally stable. Numerical simulations are computed to check the analytical expressions of adaptive controllers.

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.

A scheme is designed to achieve phase synchronization (PS) and antiphase synchronization (APS) for an n-dimensional hyperchaotic complex nonlinear system. For this scheme, we have used the idea of an active control technique based on Lyapunov stability analysis to determine analytically the complex control functions which are needed to achieve PS and APS. We applied this scheme, as an example, to study PS and APS of hyperchaotic attractors of two identical hyperchaotic complex Lorenz systems. These complex systems appear in many important fields of physics and engineering. Our scheme can also be applied to two different hyperchaotic complex systems, for which PS and APS have not been investigated, as far as we know, in the literature. Numerical results are plotted to show phases and amplitudes of these hyperchaotic attractors, thus demonstrating that PS and APS are achieved. The bifurcation diagrams are computed for a wide range of parameters of the system parameters and are found to be symmetrical

In this paper, we study the lag synchronization (LS) of n-dimensional hyperchaotic complex nonlinear systems. The idea of the nonlinear control technique based on the complex Lyapunov function with lag in time is used to propose a scheme to investigate LS of hyperchaotic attractors of these systems. Both complex Lyapunov and control functions are introduced. For illustration, the scheme is applied to two hyperchaotic complex Lorenz systems. The real and complex control functions are derived analytically to achieve LS and to show that the complex error dynamical systems are globally stable. Numerical results are calculated to test the validity of the analytical expressions of control functions to achieve LS of two identical hyperchaotic attractors.

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Dynamical systems in the real domain are currently one of the most popular areas of scientific study. A wealth of new phenomena of bifurcations and chaos has been discovered concerning the dynamics of nonlinear systems in real phase space. There is, however, a wide variety of physical problems, which, from a mathematical point of view, can be more conveniently studied using complex variables. The main advantage of introducing complex variables is the reduction of phase space dimensions by a half. In this survey, we shall focus on such classes of autonomous, parametrically excited and modulated systems of complex nonlinear oscillators. We first describe appropriate perturbation approaches, which have been specially adapted to study periodic solutions, their stability and control. The stability analysis of these fundamental periodic solutions, though local by itself, can yield considerable information about more global properties of the dynamics, since it is in the vicinity of such solutions that the largest regions of regular or chaotic motion are observed, depending on whether the periodic solution is, respectively, stable or unstable. We then summarize some recent studies on fixed points, periodic solutions, strange attractors, chaotic behavior and the problem of chaos control in systems of complex oscillators. Some important applications in physics, mechanics and engineering are mentioned. The connection with a class of complex partial differential equations, which contains such famous examples, as the nonlinear Schrödinger and Ginzburg–Landau equations is also discussed. These complex equations play an important role in many branches of physics, e.g. fluids, superconductors, plasma physics, geophysical fluids, modulated optical waves and electromagnetic fields.