A graph‐theoretic approach to stability of neutral stochastic coupled oscillators network with time‐varying delayed coupling

In this paper, a graph‐theoretic approach for checking exponential stability of the system described by neutral stochastic coupled oscillators network with time‐varying delayed coupling is obtained. Based on graph theory and Lyapunov stability theory, delay‐dependent criteria are deduced to ensure moment exponential stability and almost sure exponential stability of the addressed system, respectively. These criteria can show how coupling topology, time delays, and stochastic perturbations affect exponential stability of such oscillators network. This method may also be applied to other neutral stochastic coupled systems with time delays. Finally, numerical simulations are presented to show the effectiveness of theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Experimental reconsideration of spatio‐temporal dynamics observed in fluid‐elastic oscillator arrays from complex system viewpoint: From vibrating pipes in heat exchangers to waving plants in agricultural fields

Transition from local complexity to global spatio‐temporal dynamics in a two‐dimensional array of fluid‐elastic oscillators is examined experimentally with an apparatus comprising 90‐1000 cantilevered rods in a wind tunnel as the Reynolds number (based on rod diameter) is increased from 200 to 900. A cluster‐pattern entropy measure is introduced as a quantitative measure of local complexity. As the intensity of interaction among neighboring elements (in this case, frequency of collisions among rods) increases, a set of the elements (in this case, a rod‐array) achieves globally better‐organized behavior. On the basis of accelerometer data, the rod impact rate versus flow velocity shows a power‐law scaling relation. Video images reveal that, initially, each rod moves individually; then clusters consisting of several rods emerge. Finally, global wave‐like motion occurs at higher flow velocities. Each wave‐like motion has its specific frequency and spatial wavelength, which vary according to wind velocity. © 2007 Wiley Periodicals, Inc. Complexity 12: 36–47, 2007     

Non‐standard finite difference schemes for multi‐dimensional second‐order systems in non‐smooth mechanics

This work is an extension of the paper (Proc. R. Soc. London 2005; 461A :1927–1950) to impact oscillators with more than one degree of freedom. Given the complex and even chaotic behaviour of these non‐smooth mechanical systems, it is essential to incorporate their qualitative physical properties, such as the impact law and the frequencies of the systems, into the envisaged numerical methods if the latter is to be reliable. Based on this strategy, we design several non‐standard finite difference schemes. Apart from their excellent error bounds and unconditional stability, the schemes are analysed for their efficiency to preserve some important physical properties of the systems including, among others, the conservation of energy between consecutive impact times, the periodicity of the motion and the boundedness of the solutions. Numerical simulations that support the theory are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

New fifth‐order two‐derivative Runge‐Kutta methods with constant and frequency‐dependent coefficients

Two‐derivative Runge‐Kutta methods are Runge‐Kutta methods for problems of the form y′ = f(y) that include the second derivative y′′ = g(y) = f ′(y)f(y) and were developed in the work of Chan and Tsai. In this work, we consider explicit methods and construct a family of fifth‐order methods with three stages of the general case that use several evaluations of f and g per step. For problems with oscillatory solution and in the case that a good estimate of the dominant frequency is known, methods with frequency‐dependent coefficients are used; there are several procedures for constructing such methods. We give the general framework for the construction of methods with variable coefficients following the approach of Simos. We modify the above family to derive methods with frequency‐dependent coefficients following this approach as well as the approach given by Vanden Berghe. We provide numerical results to demonstrate the efficiency of the new methods using three test problems.     

Two‐level Fourier analysis of multigrid for higher‐order finite‐element discretizations of the Laplacian

In this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher‐order finite‐element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse‐grid correction is assumed to be an ideal operator that annihilates the low‐frequency error components and leaves the high‐frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two‐grid convergence factor. While two‐grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is “between” two‐grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher‐order finite element discretizations, including for Q₂‐Q₁ (Taylor‐Hood) elements for the Stokes equations. Finally, we present two‐grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two‐grid and multigrid convergence factors.     

R‐weak commutativity,examples, and applications

The aim of this article is to define a new contraction and its variants in non‐Archimedean Menger probabilistic metric‐spaces, and utilize them to establish the existence of a combined common fixed point illustrating with examples. We also apply our result to integral type equations, Volterra type integral equations, damped harmonic oscillators, and nonlinear matrix equations.     

Global solutions to the Navier‐Stokes‐Landau‐Lifshitz system

This paper is dedicated to the study of the Navier‐Stokes‐Landau‐Lifshitz system. We obtain the global existence of a unique solution for this system without any small conditions imposed on the third component of the initial velocity field. Our methods mainly rely upon the Fourier frequency localization and Bony's paraproduct decomposition.     

High‐order partial differential equation de‐noising method for vibration signal

A novel approach for 1D vibration signal de‐noising filter using partial differential equation (PDE) is presented. In particular, the numerical solution of higher‐order PDE is generated, and we show that it enables the amplitude‐frequency characteristic in filter to be estimated more accurately, which results in better de‐noising performance in comparison with the low‐order PDE. The de‐noising tests on different degree of artificial noise are conducted. Experimental tests have been rigorously compared with different de‐noising methods to verify the efficacy of the proposed high‐order PDE filter method. Copyright © 2014 John Wiley & Sons, Ltd.     

Scale‐free power‐laws as interaction between progress and diffusion

While scale‐free power‐laws are frequently found in social and technological systems, their authenticity, origin, and gained insights are often questioned, and rightfully so. The article presents a newly found rank‐frequency power‐law that aligns the top‐500 supercomputers according to their performance. Pursuing a cautious approach in a systematic way, we check for authenticity, evaluate several potential generative mechanisms, and ask the “so what” question. We evaluate and finally reject the applicability of well‐known potential generative mechanisms such as preferential attachment, self‐organized criticality, optimization, and random observation. Instead, the microdata suggest that an inverse relationship between exponential technological progress and exponential technology diffusion through social networks results in the identified fat‐tail distribution. This newly identified generative mechanism suggests that the supply and demand of technology (“technology push” and “demand pull”) align in exponential synchronicity, providing predictive insights into the evolution of highly uncertain technology markets. © 2013 Wiley Periodicals, Inc. Complexity 19: 56–65, 2014     

Adaptive Fourier decomposition‐based Dirac type time‐frequency distribution

The Dirac‐type time‐frequency distribution (TFD), regarded as ideal TFD, has long been desired. It, until the present time, cannot be implemented, due to the fact that there has been no appropriate representation of signals leading to such TFD. Instead, people have been developing other types of TFD, including the Wigner and the windowed Fourier transform types. This paper promotes a practical passage leading to a Dirac‐type TFD. Based on the proposed function decomposition method, viz., adaptive Fourier decomposition, we establish a rigorous and practical Dirac‐type TFD theory. We do follow the route of analytic signal representation of signals founded and developed by Garbo, Ville, Cohen, Boashash, Picinbono, and others. The difference, however, is that our treatment is theoretically throughout and rigorous. To well illustrate the new theory and the related TFD, we include several examples and experiments of which some are in comparison with the most commonly used TFDs. Copyright © 2016 John Wiley & Sons, Ltd.     

Bifurcations and spatiotemporal patterns in a ratio‐dependent diffusive Holling‐Tanner system with time delay

In this paper, we concentrate on the spatiotemporal patterns of a delayed reaction‐diffusion Holling‐Tanner model with Neumann boundary conditions. In particular, the time delay that is incorporated in the negative feedback of the predator density is considered as one of the principal factors to affect the dynamic behavior. Firstly, a global Turing bifurcation theorem for τ = 0 and a local Turing bifurcation theorem for τ > 0 are given. Then, further considering the degenerated situation, we derive the existence of Bogdanov‐Takens bifurcation and Turing‐Hopf bifurcation. The normal form method is used to study the explicit dynamics near the Turing‐Hopf singularity. It is shown that a pair of stable nonconstant steady states (stripe patterns) and a pair of stable spatially inhomogeneous periodic solutions (spot patterns) could be bifurcated from a positive equilibrium. Moreover, the Turing‐Turing‐Hopf–type spatiotemporal patterns, that is, a subharmonic phenomenon with two spatial wave numbers and one temporal frequency, are also found and explained theoretically. Our results imply that the interaction of Turing and Hopf instabilities can be considered as the simplest mechanism for the appearance of complex spatiotemporal dynamics.     

Synchronization of nonlinear master–slave systems under input delay and slope‐restricted input nonlinearity

This article addresses the synchronization of nonlinear master–slave systems under input time‐delay and slope‐restricted input nonlinearity. The input nonlinearity is transformed into linear time‐varying parameters belonging to a known range. Using the linear parameter varying (LPV) approach, applying the information of delay range, using the triple‐integral‐based Lyapunov–Krasovskii functional and utilizing the bounds on nonlinear dynamics of the nonlinear systems, nonlinear matrix inequalities for designing a simple delay‐range‐dependent state feedback control for synchronization of the drive and response systems is derived. The proposed controller synthesis condition is transformed into an equivalent but relatively simple criterion that can be solved through a recursive linear matrix inequality based approach by application of cone complementary linearization algorithm. In contrast to the conventional adaptive approaches, the proposed approach is simple in design and implementation and is capable to synchronize nonlinear oscillators under input delays in addition to the slope‐restricted nonlinearity. Further, time‐delays are treated using an advanced delay‐range‐dependent approach, which is adequate to synchronize nonlinear systems with either higher or lower delays. Furthermore, the resultant approach is applicable to the input nonlinearity, without using any adaptation law, owing to the utilization of LPV approach. A numerical example is worked out, demonstrating effectiveness of the proposed methodology in synchronization of two chaotic gyro systems. © 2015 Wiley Periodicals, Inc. Complexity 21: 220–233, 2016     

A combined filtering approach to high‐frequency volatility estimation with mixed‐type microstructure noises

This paper introduces a solution that combines the Kalman and particle filters to the challenging problem of estimating integrated volatility using high‐frequency data where the underlying prices are perturbed by a mixture of random noise and price discreteness. An explanation is presented of how the proposed combined filtering approach is able to correct for bias due to this mixed‐type microstructure effect. Simulation and empirical studies on the tick‐by‐tick trade price data for four US stocks in the year 2009 show that our method has clear advantages over existing high‐frequency volatility estimation methods.     

3‐Factor‐Criticality of Vertex‐Transitive Graphs

A graph of order n is p ‐factor‐critical, where p is an integer of the same parity as n, if the removal of any set of p vertices results in a graph with a perfect matching. 1‐factor‐critical graphs and 2‐factor‐critical graphs are factor‐critical graphs and bicritical graphs, respectively. It is well known that every connected vertex‐transitive graph of odd order is factor‐critical and every connected nonbipartite vertex‐transitive graph of even order is bicritical. In this article, we show that a simple connected vertex‐transitive graph of odd order at least five is 3‐factor‐critical if and only if it is not a cycle.     

Low‐frequency scattering by a penetrable body with an eccentric soft or hard core

A plane wave is scattered by an acoustically soft or hard sphere, covered by a penetrable non‐concentric spherical lossless shell that disturbs the propagation of the incident wave field. The dimensions of the coated sphere are much smaller than the wavelength of the incident field. Low‐frequency theory reduces this scattering problem to a sequence of potential problems, which can be solved iteratively. Exactly one bispherical coordinate system exists that fits the given geometry of the obstacle. For the case of a soft and hard core, the exact low‐frequency coefficients of the zeroth and the first‐order for the near field as well as the first‐ and second‐order coefficients for the normalized scattering amplitude are obtained and the cross sections are calculated. Discussion of the results and their physical meaning is included. Copyright © 2009 John Wiley & Sons, Ltd.     

Directional ‐matrix compression for high‐frequency problems

Standard numerical algorithms, such as the fast multipole method or ‐matrix schemes, rely on low‐rank approximations of the underlying kernel function. For high‐frequency problems, the ranks grow rapidly as the mesh is refined, and standard techniques are no longer attractive. Directional compression techniques solve this problem by using decompositions based on plane waves. Taking advantage of hierarchical relations between these waves' directions, an efficient approximation is obtained. This paper is dedicated to directional ‐matrices that employ local low‐rank approximations to handle directional representations efficiently. The key result is an algorithm that takes an arbitrary matrix and finds a quasi‐optimal approximation of this matrix as a directional ‐matrix using a prescribed block tree. The algorithm can reach any given accuracy, and the approximation requires only units of storage, where n is the matrix dimension, κ is the wave number, and k is the local rank. In particular, we have a complexity of if κ is constant and for high‐frequency problems characterized by κ²～n. Because the algorithm can be applied to arbitrary matrices, it can serve as the foundation of fast techniques for constructing preconditioners.     

Four‐terminal reducibility and projective‐planar wye‐delta‐wye‐reducible graphs

A graph is YΔY‐reducible if it can be reduced to a vertex by a sequence of series‐parallel reductions and YΔY‐transformations. Terminals are distinguished vertices, that cannot be deleted by reductions and transformations. In this article, we show that four‐terminal planar graphs are YΔY‐reducible when at least three of the vertices lie on the same face. Using this result, we characterize YΔY‐reducible projective‐planar graphs. We also consider terminals in projective‐planar graphs, and establish that graphs of crossing‐number one are YΔY‐reducible. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 83–93, 2000     

Controlling a chaotic resonator by means of dynamic track control

Josephson junction oscillators can generate chaotic signals with a wide frequency spectrum. An improved scheme of Lyapunov functions is proposed to control chaotic resonators of this type and forces them to converge to an arbitrary selected target signal. A changeable gain coefficient is introduced into the Lyapunov function, and the controllers are designed analytically. The controllers operate automatically when the output series are deviated from the target orbit synchronously. A resistive‐capacitive‐inductive‐shunted Josephson junction in chaotic parameter region is investigated in our studies, and power consumption is estimated from the dimensionless model. It is found that the power consumption of controller is dependent on the amplitude and/or angular frequency of the external target signal to be tracked. For example, larger power costs are observed when the target signal is in larger amplitude and/or angular frequency. The numerical results are consistent with the analytical discussion. © 2014 Wiley Periodicals, Inc. Complexity 21: 370–378, 2015     

GPBi‐CGstab(L): A Lanczos‐type product method unifying Bi‐CGstab(L) and GPBi‐CG

Lanczos‐type product methods (LTPMs), in which the residuals are defined by the product of stabilizing polynomials and the Bi‐CG residuals, are effective iterative solvers for large sparse nonsymmetric linear systems. Bi‐CGstab(L) and GPBi‐CG are popular LTPMs and can be viewed as two different generalizations of other typical methods, such as CGS, Bi‐CGSTAB, and Bi‐CGStab2. Bi‐CGstab(L) uses stabilizing polynomials of degree L, while GPBi‐CG uses polynomials given by a three‐term recurrence (or equivalently, a coupled two‐term recurrence) modeled after the Lanczos residual polynomials. Therefore, Bi‐CGstab(L) and GPBi‐CG have different aspects of generalization as a framework of LTPMs. In the present paper, we propose novel stabilizing polynomials, which combine the above two types of polynomials. The resulting method is referred to as GPBi‐CGstab(L). Numerical experiments demonstrate that our presented method is more effective than conventional LTPMs.     

An approach for identifying and predicting economic recessions in real‐time using time–frequency functional models

Identifying periods of recession and expansion is a challenging topic of ongoing interest with important economic and monetary policy implications. Given the current state of the global economy, significant attention has recently been devoted to identifying and forecasting economic recessions. Consequently, we introduce a novel class of Bayesian hierarchical probit models that take advantage of dimension‐reduced time–frequency representations of various market indices. The approach we propose can be viewed as a Bayesian mixed frequency data regression model, as it relates high‐frequency daily data observed over several quarters to a binary quarterly response indicating recession or expansion. More specifically, our model directly incorporates time–frequency representations of the entire high‐dimensional non‐stationary time series of daily log returns, over several quarters, as a regressor in a predictive model, while quantifying various sources of uncertainty. The necessary dimension reduction is achieved by treating the time–frequency representation (spectrogram) as an “image” and finding its empirical orthogonal functions. Subsequently, further dimension reduction is accomplished through the use of stochastic search variable selection. Overall, our dimension reduction approach provides an extremely powerful tool for feature extraction, yielding an interpretable image of features that predict recessions. The effectiveness of our model is demonstrated through out‐of‐sample identification (nowcasting) and multistep‐ahead prediction (forecasting) of economic recessions. In fact, our results provide greater than 85% and 80% out‐of‐sample forecasting accuracy for recessions and expansions respectively, even three quarters ahead. Finally, we illustrate the utility and added value of including time–frequency information from the NASDAQ index when identifying and predicting recessions. Copyright © 2012 John Wiley & Sons, Ltd.