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.

In this paper we start with two-dimensional (2D) linear cellular automata (CA) in relation with basic mathematical structure. We investigate uniform linear 2D CA over ternary field, i.e. ℤ₃. Present work is related to theoretical and imaginary investigations of 2D linear CA. Even though the basic construction of a CA is a discrete model, its macroscopic level behavior at large times and on large scales could be a close approximation to a continuous system. Considering some statistical properties, someone may also study geometrical aspects of patterns generated by cellular automaton evolution. After iteratively applying the linear rules, CA have been shown capable of producing interesting complex behaviors. Some examples of CA produce remarkably regular behavior on finite configurations. Using some simple initial configurations, the produced pattern can be self-replicating regarding some linear rules. Here we deal with the theory 2D uniform periodic, adiabatic and reflexive boundary CA (2D PB, AB and RB) over the ternary field ℤ₃ and the applications of image processing for patterns generation. From the visual appearance of the patterns, it is seen that some rules display sensitive dependence on boundary conditions and their rule numbers.

Optical bistability (OB) for a homogeneously broadened two-level atomic system in a ring cavity is investigated within and without the rotating wave approximation (RWA) using nonautonomous Maxwell–Bloch equations subject to a time delay. It is shown that the dynamics both within and without the RWA are susceptible to the introduction of time delays in the differential equations. A range of instability scenarios are found as certain parameters are ramped up and down and these can affect the bistable operation of the physical devices. However, the introduction of a time delay can also result in an important positive application; for the first time, as far as the authors are aware, it is shown that a type of butterfly hysteresis can occur in the fundamental component, which is a relatively strong output signal, and may be desired in optical signal processing.

Structural and parametric identification of nonlinear continuous dynamic systems with a closed cycle on a set of continuous block-oriented models with feedback is considered. The method of structural identification in the steady state based on the observation of the system's input and output variables at the input periodic influences is proposed. The solution of the parameter identification problems, which can be immediately connected with the structural identification problem, is carried out in the steady and transient states by the method of least squares. The structural and parametric identification algorithms are investigated by means of both theoretical analysis and computer modeling.

The paper characterizes the global threshold dynamics of an epidemic model of SIQS type in environments with fluctuations, where the quarantine class is explicitly involved. Criteria are established for the permanence and extinction of the infective in environments with time oscillations. In particular, we further consider an environment which varies periodically in time. The global threshold dynamic scenarios i.e. the existence and global asymptotic stability of the disease-free periodic solution, the existence of the endemic periodic solution and the permanence of the infective are completely characterized by the basic reproduction number defined by the spectral radius of an associated linear integral operator.

We study the symmetries of the spectrum of the Feinberg–Zee Random Hopping Matrix introduced in [J. Feinberg and A. Zee, Spectral curves of non-Hermitian Hamiltonians, Nucl. Phys. B552 (1999) 599–623] and studied in various papers thereafter (e.g. [S. N. Chandler-Wilde, R. Chonchaiya and M. Lindner, eigenvalue Problem meets Sierpinski triangle: Computing the spectrum of a non-self-adjoint random operator, Oper. Matrices5 (2011) 633–648; S. N. Chandler-Wilde, R. Chonchaiya and M. Lindner, On the spectra and pseudospectra of a class of non-self-adjoint random matrices and operators, Oper. Matrices7 (2013) 739–775; S. N. Chandler-Wilde and E. B. Davies, Spectrum of a Feinberg–Zee sandom hopping matrix, J. Spectral Theory2 (2012) 147–179; R. Hagger, On the spectrum and numerical range of tridiagonal random operators, preprint (2014), arXiv: 1407.5486; D. E. Holz, H. Orland and A. Zee, On the remarkable spectrum of a non-Hermitian random matrix model, J. Phys. A: Math. Gen.36 (2003) 3385–3400]). In [J. Spectral Theory2 (2012) 147–179], Chandler-Wilde and Davies proved that the spectrum of the Feinberg–Zee Random Hopping Matrix is invariant under taking square roots, which implied that the unit disk is contained in the spectrum (a result already obtained slightly earlier in [Oper. Matrices5 (2011) 633–648]. In a similar approach we show that there is an infinite sequence of symmetries at least in the periodic part of the spectrum (which is conjectured to be dense). Using these symmetries and the result of [J. Spectral Theory2 (2012) 147–179], we can exploit a considerably larger part of the spectrum than the unit disk. As a further consequence we find an infinite sequence of Julia sets contained in the spectrum. These facts may serve as a part of an explanation of the seemingly fractal-like behavior of the boundary.

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.

In this paper, we disscuss one-dimensional hexagonal quasicrystals and infinite elasticity plane with single periodic of second fundamental problem by the condition basical hypothesis. We use the plane elasticity of complex variables method and Hilbert kernel integral formula proving the onedimensional hexagonal quasicrystals and infinite elasticity plane with single periodic of second fundamental problem, and obtaining the solution is not only existence but also unique. And in the proving process, we provide the specific arithmetic. At last, we offer an example to verify.