Uniform asymptotic stability of impulsive discrete systems with time delay

This paper studies impulsive discrete systems with time delay. Some novel criteria on uniform asymptotic stability are established by using the method of Lyapunov functions and the Razumikhin-type technique. Examples are presented to illustrate the criteria.     

Stability of large-scale discrete systems under structural perturbations

By using the matrix Lyapunov function, we establish conditions of (uniform) stability and (uniform) asymptotic stability of a large-scale discrete system under structural perturbations.     

On the long‐time stability of the Crank–Nicolson scheme for the 2D Navier–Stokes equations

In this article we study the stability for all positive time of the Crank–Nicolson scheme for the two‐dimensional Navier–Stokes equations. More precisely, we consider the Crank–Nicolson time discretization together with a general spatial discretization, and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the numerical scheme is stable, provided a CFL‐type condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Banach空间中线性离散时间系统的一致多项式膨胀性

给出了Banach 空间中线性离散时间系统一致多项式膨胀性的概念,并讨论了其离散特征。借助Lyapunov函数给出了线性离散时间系统满足一致多项式膨胀的充要条件。所得结论将一致指数稳定性、指数膨胀性及多项式稳定性中的若干经典结论推广到了一致多项式膨胀性的情形。     

Exponential stability for linear skew-product flows

We give general characterizations for uniform exponential stability of linear skew-product flows. We present a unified treatment for discrete and integral conditions for uniform exponential stability. As applications, for the particular case of evolution families, we generalize some results due to Przyluski, Rolewicz and Zabczyk.     

Approximation of the long‐term dynamics of the dynamical system generated by the multilayer quasigeostrophic equations of the ocean

In this article, we study the multilayer quasigeostrophic equations of the ocean. More precisely, we discretize these equations in time using the implicit Euler scheme and using the classical and uniform discrete Gronwall lemmas we prove that the approximate solution is uniformly bounded in H^?1, L² and H¹. Using the uniform stability of the scheme and the theory of the multivalued attractors, we then prove that the discrete attractors generated by the numerical scheme converge to the global attractor of the continuous system as the time‐step approaches zero. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1041–1065, 2016     

Uniformly convergent difference schemes for a singularly perturbed third order boundary value problem

In this paper we consider a numerical approximation of a third order singularly perturbed boundary value problem by an upwind finite difference scheme on a Shishkin mesh. The behavior of the solution, and the stability of the continuous problem are discussed. The proof of the uniform convergence of the proposed numerical method is based on the strongly uniform stability and a weak consistency property of the discrete problem. Numerical experiments verify our theoretical results.     

ON UNIFORM ASYMPTOTIC STABILITY OF INFINITE DELAY DIFFERENCE EQUATIONS

51. IntroductionThe aim of thi8 paPer is to establish the stability criteria for the indnite de1ay differenceequations of the formx(n 1) = F(n,x.) for n E Z, (1'l)where F: Z x CH -- Rk, Z denotes the integer set, Rh is the n-dimensional Euclidean space,CH = {T E C: llWII < H} fOr some constan H > 0, whileC = {yt: {... l --2, --1,0} - Rk l W is bounded}withllytIl = sup IW(8)I for W E C,8<0and x.(8) = x(n 8) for 8 5 0. Here, and in the sequel, l' I is a norm in Rk, and we atwa…     

Mean square stability for systems on stochastically generated discrete time scales

We consider dynamic systems which evolve on discrete time domains where the time steps form a sequence of independent, identically distributed random variables. In particular, we classify the mean-square stability of linear systems on these time domains using quadratic Lyapunov functionals. In the case where the system matrix is a function of the time step, our results agree with and generalize stability results found in the Markov jump linear systems literature. In the case where the system matrix is constant, our results generalize, illuminate, and extend to the stochastic realm results in the field of dynamic equations on time scales. In order to help see the factors that contribute to stability, we prove a sufficient condition for the solvability of the Lyapunov equation by appealing to a fixed point theorem of Ran and Reurings. Finally, an example using observer-based feedback control is presented to demonstrate the utility of the results to control engineers who cannot guarantee uniform timing of the system.     

A finite difference scheme for solving the undamped Timoshenko beam equations with both ends free

In this paper, a boundary feedback system of a class of non-uniform undamped Timoshenko beam with both ends free is considered. A linearized three-level difference scheme for the Timoshenko beam equations is derived by the method of reduction of order on uniform meshes. The unique solvability, unconditional stability and convergence of the difference scheme are proved by the discrete energy method. The convergence order in maximum norm is of order two in both space and time. The validity of this theoretical analysis is verified experimentally.     

A quantitative study on the stability of delay discrete singular systems

For quadratic delay discrete singular systems, an algebraic criterion on the stability is established, and the size of the uniform stability region and asymptotic stability region around zero is estimated. Hence, the criterion is both qualitative and quantitative. With the computer techniques, the criterion dependent of delay is easy test and applies to the application in the practice. An illustrative simulation is given to illustrate the application of the obtained result.     

A numerical algorithm for simulation of the Q-switched fiber laser using the travelling wave model

We develop a finite-difference scheme for approximation of a system of nonlinear PDEs describing the Q-switching process. We construct it by using staggered grids. The transport equations are approximated along characteristics, and quadratic nonlinear functions are linearized using a special selection of staggered grids. The stability analysis proves that a connection between time and space steps arises only due to approximation requirements in order to follow exactly the directions of characteristics. The convergence analysis of this scheme is done in two steps. First, some estimates of the uniform boundedness of the discrete solution are proved. This part of the analysis is done locally, in some neighborhood of the exact solution. Second, on the basis of the obtained estimates, the main stability inequality is proved. The second-order convergence rate with respect to the space and time coordinates follows from this stability estimate. Using the obtained convergence result, we prove that the local stability analysis in the selected neighborhood of the exact solution is sufficient.     

Uniform stability and weak ergodicity of nonhomogeneous Markov chains defined on ordered Banach spaces with a base

In the present paper, we define an ergodicity coefficient of a positive mapping defined on ordered Banach space with a base , and study its properties. The defined coefficient is a generalization of the well-known the Dobrushin’s ergodicity coefficient. By means of the ergodicity coefficient we provide uniform asymptotical stability conditions for nonhomogeneous discrete Markov chains (NDMC). These results are even new in case of von Neumann algebras. Moreover, we find necessary and sufficient conditions for the weak ergodicity of NDMC. Certain relations between uniform asymptotical stability and weak ergodicity are considered.     

一类具有离散时滞的非自治扩散的两种群模型的一致持续生存

持续生存概念是种群生态系统稳定性的一个重要描述,而研究竞争种群共存的问题是种群生态学的一个重要问题.进一步考虑具有离散时滞的非自治的两种群竞争扩散摸型,使模型更符合其生态意义,通过微分不等式获得了其一致持续生存的充分条件.     

具有离散时滞的非自治扩散模型的周期解

考虑具有离散时滞及周期系数的非自治的两种群竞争扩散摸型,利用微分不等式等获得了其一致持续生存的条件,通过构造李亚普诺夫泛函获得了其正周期解存在与全局渐近稳定的充分条件.     

Nonuniform exponential contractions and Lyapunov sequences

For a nonautonomous dynamics with discrete time obtained from the product of linear operators, we show that a nonuniform exponential contraction can be completely characterized in terms of what we call strict Lyapunov sequences. We note that nonuniform exponential contractions include as a very particular case the uniform exponential contractions that correspond to have a uniform asymptotic stability of the dynamics. We also obtain “inverse theorems” that give explicitly strict Lyapunov sequences for each nonuniform exponential contraction. Essentially, the Lyapunov sequences are obtained in terms of what are usually called Lyapunov norms, that is, norms with respect to which the behavior of a nonuniform exponential contraction becomes uniform. We also show how the characterization of nonuniform exponential contractions in terms of quadratic Lyapunov sequences can be used to establish in a very simple manner the persistence of the asymptotic stability of a nonuniform exponential contraction under sufficiently small linear or nonlinear perturbations. Moreover, we describe an appropriate version of our results in the context of ergodic theory showing that the existence of an eventually strict Lyapunov function implies that all Lyapunov exponents are negative almost everywhere.     

Polynomial contractions and Lyapunov regularity

The purpose of this note is twofold: to introduce the notion of polynomial contraction for a linear nonautonomous dynamics with discrete time, and to show that it persists under sufficiently small linear and nonlinear perturbations. The notion of polynomial contraction mimics the notion of exponential contraction, but with the exponential decay replaced by a polynomial decay. We show that this behavior is exhibited by a large class of dynamics, by giving necessary conditions in terms of “polynomial” Lyapunov exponents. Finally, we establish the persistence of the asymptotic stability of a polynomial contraction under sufficiently small linear and nonlinear perturbations. We also consider the case of nonuniform polynomial contractions, for which the Lyapunov stability is not uniform.     

Razumikhin–Lyapunov functional method for the stability of impulsive switched systems with time delay

This paper investigates the uniform stability and the uniform asymptotical stability of impulsive switched systems with time delay. By employing the method of Razumikhin–Lyapunov functional, several Razumikhin-type theorems of uniform stability and uniform asymptotical stability are established, which improve some of the existing results. Several examples are also given to illustrate the results.     

Some results on cosymplectic manifolds

We obtain a generalization of the Kodaira-Morrow stability theorem for cosymplectic structures. We investigate cosymplectic geometry on Lie groups and on their compact quotients by uniform discrete subgroups. In this way we show that a compact solvmanifold admits a cosymplectic structure if and only if it is a finite quotient of a torus.