Analysis of stable two-dimensional patterns in contractile cytogel

Summary Contractile actomyosin systems play a central role in the generation of intracellular patterns. Models for pattern formation have benefited greatly from the application of mechanochemical theory. However, investigations of the patterns have been primarily qualitative in nature; the two-dimensional nature of the evolving patterns has not yet been addressed mathematically, nor has the evolution of stable heterogeneous steady-state solutions. We consider these issues, supporting our analytical predictions with numerical simulations in one and two spatial dimensions. We show how, for certain gels, the two and three-dimensional tensor equation which describes a balance of forces can be reduced to a reaction-diffusion equation.     

Global dynamics of a 3 × 6 exponential system of difference equations

We study the global dynamics of following 3×6 exponential system of difference equations: $$x_{n+1}=\frac{{\alpha }_1+{\beta }_1{e}^{?x_n}}{{\gamma }_1+y_{n?1}},y_{n+1}=\frac{{\alpha }_2+{\beta }_2{e}^{?y_n}}{{\gamma }_2+z_{n?1}},z_{n+1}=\frac{{\alpha }_3+{\beta }_3{e}^{?z_n}}{{\gamma }_3+x_{n?1}},n=0,1,\text{?},$$ where parameters α_i,β_i,γ_i (i=1,2,3) and initial conditions x_i,y_i,z_i (i=0,?1) are nonnegative real numbers. The proposed work is considerably extended and improve some existing results in the literature. Finally, theoretical results are verified by numerical simulations.     

几类Lurie型控制系统绝对稳定性充分必要条件

本借助部分变元绝对稳定性的充要条件，得出几类Lurie型控制系统绝对稳定性充分必要条件，这些判据为显式的代数形式，易于验证，也便于在实践中应用。     

On the formation of sharp transition layers in two-dimensional reaction-diffusion models

For a singularly perturbed parabolic equation in two dimensions, the formation of a solution with a sharp transition layer from a sufficiently general initial function is considered. An asymptotic analysis is used to estimate the time required for the formation of a contrast structure. Numerical results are presented.     

Multi-harmonic measurements and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions

This study presents a direct comparison of measured and predicted nonlinear vibrations of a clamped–clamped steel beam with non-ideal boundary conditions. A multi-harmonic comparison of simulations with measurements is performed in the vicinity of the primary resonance. First of all, a nonlinear analytical model of the beam is developed taking into account non-ideal boundary conditions. Three simulation methods are implemented to investigate the nonlinear behavior of the clamped–clamped beam. The method of multiple scales is used to compute an analytical expression of the frequency response which enables an easy updating of the model. Then, two numerical methods, the Harmonic Balance Method and a time-integration method with shooting algorithm, are employed and compared one with each other. The Harmonic Balance Method enables to simulate the vibrational stationary response of a nonlinear system projected on several harmonics. This study then proposes a method to compare numerical simulations with measurements of all these harmonics. A signal analysis tool is developed to extract the system harmonics’ frequency responses from the temporal signal of a swept sine experiment. An evolutionary updating algorithm (Covariance Matrix Adaptation Evolution Strategy), coupled with highly selective filters is used to identify both fundamental frequency and harmonic amplitudes in the temporal signal, at every moment. This tool enables to extract the harmonic amplitudes of the output signal as well as the input signal. The input of the Harmonic Balance Method can then be either an ideal mono-harmonic signal or a multi-harmonic experimental signal. Finally, the present work focuses on the comparison of experimental and simulated results. From experimental output harmonics and numerical simulations, it is shown that it is possible to distinguish the nonlinearities of the clamped–clamped beam and the effect of the non-ideal input signal.     

On Stability of Solutions to One Class of Nonlinear Difference Systems

We consider some class of systems of nonlinear ordinary differential equations. We adjust the difference schemes corresponding to the equations under study in order to guarantee agreement between differential and difference systems in the sense of stability of the zero solution. We obtain conditions under which perturbations do not violate the asymptotic stability of solutions to difference systems.     

Three‐point bending tests. Part II: An improved formula for the modulus of rupture and numerical simulations

The goal of this paper is to analyze analytical and numerically, from several perspectives, the modulus of rupture (MOR) for brittle materials, studying the bending test of three points which is normally used in laboratory to calculate it. In particular, we will give four different approaches to the MOR: through the classical theory of beams; by means of the one‐ and three‐dimensional numerical simulations; and by using an improved expression to the MOR obtained through its asymptotic analysis. Finally, we will present these methodologies for cylindrical and rectangular beams made of porcelain. Copyright © 2011 John Wiley & Sons, Ltd.     

A class of one-parameter alternating direction implicit methods for two-dimensional wave equations with discrete and distributed time-variable delays

For solving the initial-boundary value problem of two-dimensional wave equations with discrete and distributed time-variable delays, in the present paper, we first construct a class of basic one-parameter methods. In order to raise the computational efficiency of this class methods, we remold the methods as one-parameter alternating direction implicit (ADI) methods. Under the suitable conditions, the remolded methods are proved to be stable and convergent of second order in both of time and space. With several numerical experiments, the computational effectiveness and theoretical exactness of the methods are confirmed. Moreover, it is illustrated that the proposed one-parameter ADI method has the better advantage in computational efficiency than the basic one-parameter methods.     

Numerical stability analysis of the Taylor-Couette flow in the two-dimensional case

The stability of the laminar flow between two rotating cylinders (Taylor-Couette flow) is numerically studied. The simulation is based on the equations of motion of an inviscid fluid (Euler equations). The influence exerted on the flow stability by physical parameters of the problem (such as the gap width between the cylinders, the initial perturbation, and the velocity difference between the cylinders) is analyzed. It is shown that the onset of turbulence is accompanied by the formation of large vortices. The results are analyzed and compared with those of similar studies.     

STABILITY OF DIFFERENCE SCHEMES FOR NONLINEAR PARABOLIC SYSTEMS WITH THREE DIMENSIONS

1.Introduction1).Inthepresentworkwearegoingtostudythedeferenceschemesfortheboundaryvalueproblemofthenonlinearparabolicsystemsofpartialdifferentialequationswithtwoandthreespacedimensions.Inthecaseofthreespacedimension,thenonlinearparabolicsystemisoftheformat~A(x,y,z,t,u,aamactu.)(u.. aam u..) f(x,y,z,t,aliojac,u.),(1)whereu=(altuZI',urn)istheunknownm-dimensionalvectorfunction(m3l),A(x,yiitt,u,PI,pz,ps)isagivenmXmmatrixfunction,f(x,y,itt,utpl,p2,p3)isthegivenmdimensionalvectorfunctionandalso…     

Stability analysis of numerical methods for systems of neutral delay-differential equations

Stability analysis of some representative numerical methods for systems of neutral delay-differential equations (NDDEs) is considered. After the establishment of a sufficient condition of asymptotic stability for linear NDDEs, the stability regions of linear multistep, explicit Runge-Kutta and implicitA-stable Runge-Kutta methods are discussed when they are applied to asymptotically stable linear NDDEs. Some mentioning about the extension of the results for the multiple delay case is given.     

Stability of periodic traveling flexural‐gravity waves in two dimensions

In this work, we solve the Euler's equations for periodic waves traveling under a sheet of ice. These waves are referred to as flexural‐gravity waves. We compare and contrast two models for the effect of the ice: a linear model and a nonlinear model. The benefit of this reformulation is that it facilitates the asymptotic analysis. We use it to derive the nonlinear Schrödinger equation that describes the modulational instability of periodic traveling waves. We compare this asymptotic result with the numerical computation of stability using the Fourier–Floquet–Hill method to show they agree qualitatively. We show that different models have different stability regimes for large values of the flexural rigidity parameter. Numerical computations are also used to analyze high‐frequency instabilities in addition to the modulational instability. In the regions examined, these are shown to be the same regardless of the model representing ice.     

一类高阶时滞差分方程的有界持久性与全局渐近稳定性

获得一类高阶时滞差分方程解的有界持久性和全局渐近稳定性的充分条件;所得结果部分地解决了G.Ladas的2个公开问题,推广了一些已知结果。     

The Exact Limits and Improved Decay Estimates for All Order Derivatives of the Global Weak Solutions to a Two-Dimensional Incompressible Dissipative Quasi-Geostrophic Equation

We will accomplish the exact limits for all order derivatives of the global weak solutions to a two-dimensional incompressible dissipative quasi-geostrophic equation. We will also establish the improved decay estimates with sharp rates for all order derivatives. We will consider two cases for the initial function and the external force and prove the optimal results for both cases. We will couple together existing ideas (including the Fourier transformation and its properties, Parseval''s identity, iteration technique, Lebesgue''s dominated convergence theorem, Gagliardo-Nirenberg-Sobolev interpolation inequality, squeeze theorem, Cauchy-Schwartz''s inequality, etc) existing results (the existence of global weak solutions, the existence of local smooth solution on $(T,\infty)$ and the elementary decay estimate with a sharp rate) and a few novel ideas to obtain the main results.     

Preservation of persistence and stability under intersections and operations,part 2: Stability

The stability of the image of a moving convex set by a multifunction is considered. This study completes the persistence results obtained in Part 1 of this paper (Ref. 1). The results can be applied to epigraphs, hence to the convergence of convex functions.     

线性微分差分系统渐近稳定性的充要条件

本文引进了一种可调参数的方法，应用该方法研究了线性微分差分系统的渐近稳定性，得到了该系统渐近稳定性的充要条件的代数判别准则．     

Mathematical analysis and numerical simulation of chaotic noninteger order differential systems with Riemann‐Liouville derivative

This article deals with the design, analysis, and implementation of a robust numerical scheme when applied to time‐fractional reaction‐diffusion system. Stability analysis and numerical treatment of chaotic fractional differential system in Riemann‐Liouville sense are considered in this article. Simulation results show that chaotic phenomena can only occur if the reaction or local dynamics of such system is coupled or nonlinear in nature. Illustrative examples that are still of current and recurring interest to economists, engineers, mathematicians and physicists are chosen, to describe the points and queries that may arise. Numerical results presented agree with the theoretical findings.     

Numerical Approximation Of The Phase-Field Transition System With Non-Homogeneous Cauchy-Neumann Boundary Conditions In Both Unknown Functions Via Fractional Steps Method

The paper concerns with the proof of the convergence for an iterative scheme of fractional steps type associated to the phase-field transition system endowed with non-homogeneous Cauchy-Neumann boundary conditions, in both unknown functions. The advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of nonlinear parabolic system. On the basis of this approach, a numerical algorithm in 2D case is introduced and an industrial implementation is made.