Narrow Results

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 patterns in chains of N bi-directionally delayed coupled systems with delayed feedback are studied. Each system is hyperchaotic when decoupled from the chain. It is shown that chains with odd or even number of cites N display different spatial patterns of stable exact synchronization. When N is odd the only stable pattern of exact synchronization is between all of the units. When N is even, next to the nearest neighbors could become exactly synchronized, with the dynamics of the nearest neighbors related in a more complicated way. Sufficiently strong coupling leads to the nearest neighbor synchronization also for even N. No other patterns have been observed.

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 one based on Jumarie’s modified Riemann–Liouville sense. Many systems in the interdisciplinary fields could be described by fractional-order nonlinear dynamical systems, such as viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, heat conduction, resistance-capacitance-inductance (RLC) interconnect and electromagnetic waves. To deal with integer order dynamical system it would be much easier in contrast with fractional-order system. Two systems are considered as examples to illustrate the validity and advantages of this technique. We have calculated the Lyapunov exponents of these examples before and after the transformation and obtained the same conclusions. We used the integer version of our example to compute numerically the values of the fractional-order and the system parameters at which chaotic and hyperchaotic behaviors exist.

Based on the study on Jerk chaotic system, a multiscroll hyperchaotic system with hidden attractors is proposed in this paper, which has infinite number of equilibriums. The chaotic system can generate $N+M+2$ scroll hyperchaotic hidden attractors. The dynamic characteristics of the multiscroll hyperchaotic system with hidden attractors are analyzed by means of dynamic analysis methods such as Lyapunov exponents and bifurcation diagram. In addition, we have studied the synchronization of the system by applying an adaptive control method. The hardware experiment of the proposed multiscroll hyperchaotic system with hidden attractors is carried out using discrete components. The hardware experimental results are consistent with the numerical simulation results of MATLAB and the theoretical analysis results.

This work focused on introducing a new two-dimensional (2D) chaotic system. It is combined of some existing maps, the logistic, iterative chaotic map with infinite collapse, and Henon maps; we called it 2D-LCHM. The assessment of the actual performance of 2D-LCHM presents its sensitivity to a tiny change in the initial condition. Besides, its dynamics behavior is very complicated. It also has hyperchaotic properties and good ergodicity. The proposed system is occupied with designing a new image encryption system. Changing the image pixels’ locations is the primary step in the encryption process. The original image is splitting into four blocks to create four different keys based on 2D-LCHM; this reduces the computation time and increases the complexity. To obtain the encryption image, we have to repeat the partitioning process several times for each block. The encryption image’s efficiency is shown through some performance analysis such as; image histogram, the number of pixels changes rate (NPCR), the unified average changing intensity (UACI), pixels correlation, and entropy. The proposed system is compared with some efficient encryption algorithms in terms of chaocity attributes and image performance; the analysis result showed consistent improvement.

The aim of this paper is to introduce the new hyperchaotic complex Lü system. This system has complex nonlinear behavior which is studied and investigated in this work. Numerically the range of parameters values of the system at which hyperchaotic attractors exist is calculated. This new system has a whole circle of equilibria and three isolated fixed points, while the real counterpart has only three isolated ones. The stability analysis of the trivial fixed point is studied. Its dynamics is more rich in the sense that our system exhibits both chaotic and hyperchaotic attractors as well as periodic and quasi-periodic solutions and solutions that approach fixed points.

Dynamical systems described by real and complex variables are currently one of the most popular areas of scientific research. These systems play an important role in several fields of physics, engineering, and computer sciences, for example, laser systems, control (or chaos suppression), secure communications, and information science. Dynamical basic properties, chaos (hyperchaos) synchronization, chaos control, and generating hyperchaotic behavior of these systems are briefly summarized. The main advantage of introducing complex variables is the reduction of phase space dimensions by a half. They are also used to describe and simulate the physics of detuned laser and thermal convection of liquid flows, where the electric field and the atomic polarization amplitudes are both complex. Clearly, if the variables of the system are complex the equations involve twice as many variables and control parameters, thus making it that much harder for a hostile agent to intercept and decipher the coded message. Chaotic and hyperchaotic complex systems are stated as examples. Finally there are many open problems in the study of chaotic and hyperchaotic complex nonlinear dynamical systems, which need further investigations. Some of these open problems are given.