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Emergence of synchronization is a remarkable collective phenomena between apparently independent agents in numerous multilevel and complex systems. The evidence of synchronization ranges from the elementary biological organisms to the most sophisticated human societies. In this paper, the problem of synchronization of nonlinearly coupled dynamical networks of Hindmarsh–Rose neurons with a sigmoidal coupling function is addressed. Sufficient condition for synchrony in terms of network structure is developed. A study on the basis of attraction of the complete synchronization is carried out for different structured networks. Also the phase synchronization of dynamical network of Hindmarsh–Rose neurons are studied. The impact of different structural properties of complex network on the phase synchronization are analyzed. The synchronization of Hindmarsh–Rose neurons are evaluated and compared on different structured network like random, regular, small-world, scale-free and modular networks. Interestingly, it was found that networks with high clustering coefficient and neutral degree mixing pattern promote better synchronization. Some chimera like state are also found in different structural networks. Further the effect of time delay dynamics on the synchronization of nonlinearly coupled network of Hindmarsh–Rose neurons are illustrated. [ABSTRACT FROM AUTHOR]

In this paper we present a singular value decomposition for any n × n + 1 complex upper bidiagonal Toeplitz matrix. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the composite multiscale cross-sample entropy (CMCSE) which may induce undefined entropies and then further propose the refined composite multiscale cross-sample entropy (RCMCSE) which modifies CMCSE. First, we apply multiscale cross-sample entropy (MCSE), CMCSE and RCMCSE methods to three types of artificial time series in order to test the validity and accuracy of these methods. Results show that RCMCSE reduces not only standard deviation, but also the probability of inducing undefined entropy effectively, which can provide better robustness and more accurate entropies. Then, these three methods are employed to investigate financial time series including US and Chinese stock indices. For the study between stock indices in the same region, some conclusions which are consistent with previous study are drawn by the RCMCSE results. Meanwhile, it can be found that undefined entropies are induced and the numbers of inducing undefined entropy by three methods for investigation between three US stock indices and two Chinese mainland stock indices are given. Compared with MCSE and CMCSE, RCMCSE method is capable of reducing the number of undefined entropy and providing more accurate entropies. Moreover, the differences on inducing undefined entropy between results for US stock indices & two Chinese mainland stock indices and results for US stock indices & HSI demonstrate a much closer relation between US stock markets and HSI than between US stock markets and two Chinese mainland stock markets. Hence, it can be concluded that RCMCSE is more applicable for the study between US and Chinese stock markets. [ABSTRACT FROM AUTHOR]

In the last three decades a growing interest in developing nonlinear dynamical systems for economic models, displaying chaotic behavior has been developed. To this direction, a novel 3-D nonlinear finance chaotic system consisting of two nonlinearities is presented. The dynamical analysis of the proposed system confirms its complex dynamic behavior, which is studied by using well-known simulation tools of nonlinear theory, such as the bifurcation diagram, Lyapunov exponents and phase portraits. Also, some interesting phenomena related with nonlinear theory are observed, such as route to chaos through a period doubling sequence, antimonotonicity and crisis phenomena. In addition, an interesting scheme of adaptive control of finance system's behavior is presented. Furthermore, the novel nonlinear finance system is emulated by an electronic circuit and its dynamical behavior is studied by using the electronic simulation package Cadence OrCAD in order to confirm the feasibility of the theoretical model. [ABSTRACT FROM AUTHOR]

Let D be the open unit disk in the complex plane C and H ( D ) the class of all analytic functions on D . Let φ be an analytic self-map of D and ψ ∈ H ( D ) . In this paper the boundedness and compactness of the generalized weighted composition operator D φ , ψ n f = ψ f ( n ) ∘ φ from the Zygmund space to the Bloch–Orlicz space and the little Bloch–Orlicz space are characterized. [ABSTRACT FROM AUTHOR]

The complex dynamics on the single population model with impulsively unilateral diffusion between two patches was studied in a theoretical way. The existence, uniqueness and stability of an order-1 periodic solution was investigated for state-dependent impulsively differential equations. The sufficient conditions for the existence and stability of positive periodic solution were obtained using the Poincare map by comparison with the analysis for limit cycles of continuous systems, which was different from the analogue of Poincare criterion. Meanwhile, the uniqueness of periodic solution was proofed by the monotone of successor function. [ABSTRACT FROM AUTHOR]

This paper is concerned with the fast, accurate and validated evaluation of elementary symmetric functions in floating-point arithmetic. We present two new compensated algorithms, with real and complex floating-point inputs respectively, by applying error-free transformations to improve the accuracy of the so-called summation algorithm that is used, by example, in the MATLAB’s poly function. We derive forward roundoff error bounds and running error bounds for our new algorithms. The roundoff error bounds imply that the computed results are as accurate as if computed with twice the working precision and then rounded to the current working precision. The running error analysis provides a shaper bound along with the result, without increasing significantly the computational cost. Numerical experiments illustrate that our algorithms run much faster than the algorithms using the classic double–double library while sharing similar error estimates. Such algorithms can be widely applicable for example to compute characteristic polynomials from eigenvalues or polynomial’s coefficients from zeros. Some simple applications are presented to show that the proposed algorithms compute the coefficients of the characteristic polynomials of some real and complex matrices to high relative accuracy. [ABSTRACT FROM AUTHOR]

In this paper we deal with the study of non-repairable complex system which consists of two subsystems say A and B, connected in series. The subsystems A and B are weighted k -out-of- n : G and weighted l -out-of- m : G configurations respectively. The subsystem A consists of n type-L components of linear ( v, f, e ): G configuration whereas the subsystem B consists of m type- S component of circular ( v, f, e ): G configuration. All the components of the subsystems A and B are arranged in parallel. In this study evaluation of the reliability characteristics such as reliability, mean time to failure and sensitivity of the considered system with the application of universal generating function is presented. To demonstrate the model a numerical example have been taken at last in which probability of success of the components of the component of the subsystems A and B follows exponential distribution. Reliability of the system after imposing some conditions on weight has also been compared. [ABSTRACT FROM AUTHOR]

The fractional-order total variation(TV) image denoising model has been proved to be able to avoid the “blocky effect”. However, it is difficult to be solved due to the non-differentiability of the fractional-order TV regularization term. In this paper, the proximity algorithm is used to solve the fractional-order TV optimization problem, which provides an effective tool for the study of the fractional-order TV denoising model. In this method, the complex fractional-order TV optimization problem is solved by using a sequence of simpler proximity operators, and therefore it is effective to deal with the problem of algorithm implementation. The final numerical procedure is given for image denoising, and the experimental results verify the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

The nonlinear propagation of dust-acoustic (DA) waves in a magnetized dusty plasma with a pair of trapped ions is investigated. Starting from a set of hydrodynamic equations for dust fluids as well as kinetic Vlasov equations for ions, and applying the reductive perturbation technique, a Korteweg-de Vries (KdV)-like equation with a complex coefficient of nonlinearity is derived, which governs the evolution of small-amplitude DA waves in plasmas. The complex coefficient arises due to vortex-like distributions of both positive and negative ions. An analytical as well as numerical solution of the KdV equation are obtained and analyzed with the effects of external magnetic field, the dust pressure as well as different mass and temperatures of ions. [ABSTRACT FROM AUTHOR]

Recent papers in physics literature have introduced coupled Ginzburg–Landau models for complex vector-valued order parameters in order to account for ferromagnetic or antiferromagnetic effects in high-temperature superconductors. In this paper, we study the spatial behavior of interacting components of coupled Ginzburg–Landau equations. We prove that the interspecies interaction leads to extinction in three dimensions. [ABSTRACT FROM AUTHOR]

In previous years, Bai and Wang presented a class of parameterized inexact Uzawa (PIU) methods for solving the generalized saddle point problems. In this paper, we consider the same method for iteratively solving the complex symmetric linear systems. Our main contribution is accelerating the convergence of the parameterized inexact Uzawa method by correction technique. First, the corrected model for the PIU method is established and the corrected PIU method is presented. Then we study the convergence property of the corrected PIU method. In fact, the corrected PIU method can converge faster than some Uzawa-type and HSS-like methods. Finally, numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the new method. [ABSTRACT FROM AUTHOR]

In this paper, the global exponential stability of complex-valued impulsive systems is addressed. Some new sufficient conditions are obtained to guarantee the global exponential stability by the Lyapunov–Razumikhin theory, which extend and improve most of recent results. Moreover, the obtained Razumikhin conditions are very simple and efficient to verify in real problems and helpful to investigate the stability of delayed neural networks and synchronization problems of chaotic systems under impulsive perturbation. Finally, a numerical example is given to show the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

Moments and moment invariants have been widely used as a basic feature descriptors in image analysis, pattern recognition, and image retrieval. However, they are mainly used to deal with the binary or gray-scale images, which lose some significant color information. Recently, quaternion techniques were introduced to conventional image moments (including Fourier–Mellin moments, Zernike/Pseudo Zernike moments, and Bessel–Fourier moments, etc.) for describing color images, and some quaternion moment and moment invariants were developed. But, the conventional image moments usually cannot effectively capture the image information, especially the edges. Besides, the kernel computation of them involves computation of a number of factorial terms, which inevitably cause the numerical stability of these moments. Based on effective polar complex exponential transform (PCET) and algebra of quaternions, we introduced the quaternion polar complex exponential transform (QPCET) for describing color images in this paper, which can be seen as the generalization of PCET for gray-level images. It is shown that the QPCETs can be obtained from the PCET of each color channel. We derived and analyzed the rotation, scaling, and translation (RST) invariant property of QPCET. We also discussed the problem of color image retrieval using QPCET. Experimental results are provided to illustrate the efficiency of the proposed color image descriptors. [ABSTRACT FROM AUTHOR]

This paper is concerned with the boundedness of solutions of the complex Lorenz system. We have obtained the global exponential attractive set Ψλ,m and the ultimate bound Ωλ,m for this system. Furthermore, we confirm that the rate of the trajectories of the system going from the exterior of the set Ψλ,m to the interior of the set Ψλ,m is an exponential rate. The rate of the trajectories is also obtained. Numerical simulations are presented to show the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

Abstract: Using a new fixed point relation, the interval methods for the simultaneous inclusion of complex multiple zeros in circular complex arithmetic are constructed. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis for the total-step and the single-step methods with Schröder’s and Halley-like corrections under computationally verifiable initial conditions. The suggested algorithms possess a great computational efficiency since the increase of the convergence rate is attained without additional calculations. Two numerical examples are given to demonstrate convergence characteristics of the proposed method. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the positive definite solutions of the nonlinear matrix equation (NME) where A is a given complex matrix. We show that such an NME has a positive definite solution if and only if an auxiliary standard NME in the form of has. Furthermore, the relationship regarding the maximal positive definite solution and the minimal positive definite solution between these two NMEs are established. Three iterative algorithms are proposed to solve the considered NME. The first method is directly related to the considered original equation but involves matrix inversion; the second method avoids the computation of matrix inversion; while the third method is an accelerated version of the first one. Convergence performances of these three kinds of algorithms are analyzed in detail. Numerical examples are worked out to validate the effectiveness of the proposed algorithms. [Copyright &y& Elsevier]

Highlights: [•] We propose the concept of modified complex projective synchronization. [•] We consider real parts, imaginary parts and module-phase synchronization. [•] State variables and projective factors which are plurals are taken into consideration. [•] More than one lyapunov exponents are more complex. [Copyright &y& Elsevier]

Abstract: This paper deals with the stability of Runge–Kutta methods applied to the complex linear system . The condition under which the numerical solution is asymptotically stable is presented, which is stronger than A-stability and weaker than -stability. Furthermore, in the case of 2-norm and L being a real symmetric matrix, by using Pad approximation and order star theory, it is proved that for A-stable Runge–Kutta methods, suppose whose stability function is given by the -Pad approximation to , the numerical solution is asymptotically stable if and only if r is even. [Copyright &y& Elsevier]

Abstract: In recent literature, researchers have generated Julia sets and Mandelbrot sets in Mann and Ishikawa orbits that are examples of two-step and three-step feedback processes respectively. This paper presents further generalization of Julia and Mandelbrot sets for complex-valued polynomials such as quadratic, cubic and higher degree polynomials using a Noor orbit, which is a four-step iterative procedure. The graphical images of Julia and Mandelbrot sets have been visualized and certain patterns in Mandelbrot sets have been recognized. It is fascinating to see that a few Mandelbrot sets are akin to a butterfly or a coupled urn or a coupled trident. [Copyright &y& Elsevier]

Abstract: Company ranking is a complex process in which multiple financial ratios are required to be considered simultaneously. Furthermore, the selection process of an appropriate credit applicant company has become more complex as the experts in the financial organizations have to assess a wide range of alternatives based on a set of conflicting financial criteria. This paper studies the application of a new approach, i.e., Design of Experiment and TOPSIS method (DoE–TOPSIS) together to make company raking as frequently encountered in the real-time financial environment. The developed model is tested by case studies and satisfactory results are obtained. In the case studies, the results obtained by using the combined DoE–TOPSIS method are almost corroborated by those derived from conventional multi attribute decision making (MADM) methods which prove the applicability, potentiality, simplicity and flexibility of this method in making company ranking. [Copyright &y& Elsevier]

Abstract: The simulation of complex systems is important in many fields of science and in real-world applications. Such systems are composed of many interacting subsystems. There might exist different software packages for simulating the individual subsystems and co-simulation refers to the simultaneous execution of multiple interacting subsystem simulators. Simulation or co-simulation, if not designed properly, can return misleading numerical solutions (unstable numerical solutions for what is in fact a stable system or vice versa). To understand the cause of these numerical artifacts, we first propose a simple mathematical model for co-simulation, and then construct stability charts. These charts shed light on transitions between stable and unstable behavior in co-simulation. Our goal is to understand the stability properties of the simulated and co-simulated representation of the continuous system. We will achieve this goal by expressing the trace and determinant of the discretized system in terms of the trace and determinant of the continuous system to establish stability criteria. [Copyright &y& Elsevier]

Abstract: In this paper, the authors derive several sharp coefficient estimates for the subclasses of the class of normalized starlike functions of complex order . The results presented here would generalize as well as improve the recent work by Srivastava et al. (2011) [20]. Some closely-related open problems are also posed. [Copyright &y& Elsevier]

Highlights: [•] Discrete time Prey–predator system with prey refuge is considered here. [•] Predator’s attack rate largely varies with the escaping ability of the prey. [•] System experiences complex dynamics with varying the strength of prey refuge. [•] System experiences Hopf and flip bifurcations. [•] The computations of Lyapunov exponent confirms the chaotic dynamics of the system. [Copyright &y& Elsevier]

Abstract: In the classical model for portfolio selection the risk is measured by the variance of returns. It is well known that, if returns are not elliptically distributed, this may cause inaccurate investment decisions. To address this issue, several alternative measures of risk have been proposed. In this contribution we focus on a class of measures that uses information contained both in lower and in upper tail of the distribution of the returns. We consider a nonlinear mixed-integer portfolio selection model which takes into account several constraints used in fund management practice. The latter problem is NP-hard in general, and exact algorithms for its minimization, which are both effective and efficient, are still sought at present. Thus, to approximately solve this model we experience the heuristics Particle Swarm Optimization (PSO). Since PSO was originally conceived for unconstrained global optimization problems, we apply it to a novel reformulation of our mixed-integer model, where a standard exact penalty function is introduced. [Copyright &y& Elsevier]

Highlights: [•] We identified a class of integrable nonlinear complex ODEs. [•] Two types of nonlocal transformations which connect the nonlinear complex ODE with linear ODE are presented. [•] General solution of a class of nonlinear complex ODEs are given. [Copyright &y& Elsevier]

Abstract: This paper proposes a high gain integral controller for synchronization of complex dynamical networks with unknown non-identical nodes. The integral controller converts the complex dynamical system into a singular perturbation form. Then, a sufficient condition for global synchronization is derived. Finally, the numerical simulation is presented to illustrate the effectiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: In this paper, we study numerically quantized vortex dynamics and their interaction in the two-dimensional complex Ginzburg–Landau equation (CGLE) with a dimensionless parameter on bounded domains under either Dirichlet or homogeneous Neumann boundary condition. We begin with a review of the reduced dynamical laws (RDLs) for time evolution of quantized vortex centers in CGLE and show how to solve these nonlinear ordinary differential equations numerically. Then we present efficient and accurate numerical methods for solving the CGLE on either a rectangular or a disk domain under either Dirichlet or homogeneous Neumann boundary condition. Based on these efficient and accurate numerical methods for CGLE and the RDLs, we explore rich and complicated quantized vortex dynamics and interaction of CGLE with different ε and under different initial physical setups, including single vortex, vortex pair, vortex dipole and vortex lattice, compare them with those obtained from the corresponding RDLs, and identify the cases where the RDLs agree qualitatively and/or quantitatively as well as fail to agree with those from CGLE on vortex interaction. Finally, we also obtain numerically different patterns of the steady states for quantized vortex lattices in the CGLE dynamics on bounded domains. [Copyright &y& Elsevier]

Abstract: This paper examined the spectral properties of quadruple symmetric real functions. Theorem is formulated, under which the complex spectra of such symmetric functions are conjugated. The properties of the trigonometric functions are used in the proof of the theorem. Numerical results are presented for illustrating of the proven theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we study the exact solutions of ac-driven complex Ginzburg–Landau equation under the given parameter conditions. We apply the method of dynamical systems to analyze the dynamical behavior of the stationary solutions and their bifurcations depending on the parameters of systems. All bounded exact solutions are obtained. To guarantee the existence of those solutions, the constraint parameter conditions are given. [Copyright &y& Elsevier]

Abstract: By using the complex representation of quaternion matrices, and the Moore–Penrose generalized inverse, we derive the expressions of the least squares solution with the least norm, the least squares pure imaginary solution with the least norm, and the least squares real solution with the least norm for the quaternion matrix equation , respectively. Finally, we discuss their applications in color image restoration. [Copyright &y& Elsevier]

Highlights: [•] Vibration analysis is achieved by the least cost of analysis and high accuracy of results. [•] The stability and accuracy of results have been evaluated, numerically. [•] We have examined a broad-frequency-content excitation governing on a MDOF structure. [•] We have examined efficiency of the proposed scheme on a large-scaled MDOF structure. [•] A complex and a simple wavelet functions have been considered, comparatively. [ABSTRACT FROM AUTHOR]

Abstract: The complex potential theory of two-dimensional decagonal quasicrystals is constructed and the complex variable method of Muskhelishvili is developed. Based on the complex representation of stresses and displacements, the arbitrariness and restrictions on the complex potentials are discussed. As an illustration of the complex potential theory, a decagonal quasicrystals plate with an elliptic notch under the action of stretch is considered. Some special cases of the results are also observed, which are helpful to check the correctness of the complex potential theory. [Copyright &y& Elsevier]

Abstract: Let denote the class of analytic functions in the unit disc on the complex plane . If the operator satisfiesfor all , then it is called subordination-preserving operator on the class . In this paper we present a new result of this form which generalize some of the earlier and is connected with the Wilf’s conjecture. The applications of main result are also presented. [Copyright &y& Elsevier]

Abstract: The weighted Moore–Penrose inverse of a matrix can be used to define a partial order on the set of complex matrices and to introduce the concept of weighted-EP matrices. In this paper we study the weighted star partial order on the set of weighted-EP matrices. In addition, some properties that relate the eigenprojection at zero with the weighted star partial order are obtained. [Copyright &y& Elsevier]

Abstract: The synchronization for a class of complex dynamical network with similar nodes and coupling time-delay is investigated. The dimensions of the dynamical nodes in the complex network are different. Moreover, the coupling time-delays of the dynamical nodes are also different, and the decentralized control strategies are designed in term of the information of the similar parameters in dynamical complex networks by using the method of linear matrix inequality. The definition of the synchronization manifold is constructed, and the time delay independent criterion of the synchronization of the complex dynamical networks is derived. Finally, a numerical example is presented to demonstrate the application of the theoretical results. [Copyright &y& Elsevier]

Abstract: The dynamical behavior of two iterative derivative-free schemes, Steffensen and M4 methods, is studied in case of quadratic and cubic polynomials. The parameter plane is analyzed for both procedures on quadratic polynomials. Different dynamical planes are showed when the mentioned methods are applied on particular cubic polynomials with real or complex coefficients. The property of immersion of the basins of attraction in all cases is analyzed. [Copyright &y& Elsevier]

Abstract: Based on the basic equations of plane and anti-plane problems and the feature of stress fields, four important properties for complex potentials expressed in power series expansion in the plane and anti-plane elastic fields are obtained. First, the complex potentials in the plane or anti-plane elastic fields are both even functions when the stress states are symmetrical with regard to the origin. Next, it is found that the coefficients of complex potentials in the plane elastic field must be real quantity when the stress states are symmetrical with regard to the x-axis. The last conclusion is that the complex potentials in the anti-plane elastic field only have imaginary coefficients when the stress states are anti-symmetric with regard to the x-axis. Then, based on the above conclusions some classic solutions are rearrived at, which indicates the results derived in the paper are right and may simplify the process of constructing and solving the complex potential functions. The present conclusions provide an efficient tool to discover the complex potentials in the sophisticated state. [Copyright &y& Elsevier]