New explicit global asymptotic stability criteria for higher order difference equations

New explicit sufficient conditions for the asymptotic stability of the zero solution of higher order difference equations are obtained. These criteria can be applied to autonomous and nonautonomous equations. The celebrated Clark asymptotic stability criterion is improved. Also, applications to models from mathematical biology and macroeconomics are given.     

Some results on linear nabla Riemann-Liouville fractional difference equations

In this paper, we establish some criteria for boundedness, stability properties, and separation of solutions of autonomous nonlinear nabla Riemann-Liouville scalar fractional difference equations. To derive these results, we prove the variation of constants formula for nabla Riemann-Liouville fractional difference equations.     

线性随机微分方程多步法的稳定性

研究了多步法用于求解线性随机微分方程的稳定性,利用维纳过程的增量服从正态分布的性质,得到了在乘性噪声情况下,多步法用于线性随机微分方程的均方稳定性的条件,并用MATLAB对实际算例进行了数值模拟.     

Invariance Principles for Autonomous Infinite Delay Difference Systems

The invariance principles for autonomous difference systems with infinite delay are established.As applications of the obtained invariance principles, criteria for asymptotic stability and asymptotic constancy of solutions are also given.     

Stability of solutions to differential equations with several variable delays. I

We consider a class of scalar linear differential equations with several variable delays and constant coefficients. A family of equations of the class is defined by coefficients and maximum admissible values of delays. We obtain conditions that are necessary and sufficient for the stability of solutions to all equations of the family. It is ascertained that the conditions are determined entirely by properties of the solution to the initial problem for an autonomous equation that belongs to the family. Some alternatives of required conditions are obtained in the form of estimates for solutions to autonomous equations in a finite interval.     

Critical points at infinity and blow up of solutions of autonomous polynomial differential systems via compactification

Critical points at infinity for autonomous differential systems are defined and used as an essential tool. Rn is mapped onto the unit ball by various mappings and the boundary points of the ball are used to distinguish between different directions at infinity. These mappings are special cases of compactifications. It is proved that the definition of the critical points at infinity is independent of the choice of the mapping to the unit ball.We study the rate of blow up of solutions in autonomous polynomial differential systems of equations via compactification methods. To this end we represent each solution as a quotient of a vector valued function (which is a solution of an associated autonomous system) by a scalar function (which is a solution of a related scalar equation).     

Uniformly attracting solutions of nonautonomous differential equations

Understanding the structure of attractors is fundamental in nonautonomous stability and bifurcation theory. By means of clarifying theorems and carefully designed examples we highlight the potential complexity of attractors for nonautonomous differential equations that are as close to autonomous equations as possible. We introduce and study bounded uniform attractors and repellors for nonautonomous scalar differential equations, in particular for asymptotically autonomous, polynomial, and periodic equations. Our results suggest that uniformly attracting or repelling solutions are the true analogues of attracting or repelling fixed points of autonomous systems. We provide sharp conditions for the autonomous structure to break up and give way to a bewildering diversity of nonautonomous bifurcations.     

广义Birkhoff自治系统的平衡稳定性

研究广义Ｂｉｒｋｈｏｆ自治系统的平衡稳定性问题·首先建立了广义Ｂｉｒｋｈｏｆ自治系统的平衡方程,然后研究平衡状态稳定性的一次近似方法和直接法,并应用Ляпунов定理得到了广义Ｂｉｒｋｈｏｆ自治系统平衡稳定性的一些结果·最后举例说明了这些结果的应用     

一类输出耦合混沌复杂动态网络不稳定平衡点的控制

研究节点输出耦合混沌复杂动态网络不稳定平衡点的控制问题,基于输出控制思想,提出网络节点不稳定平衡点的全局控制方法以及牵制控制方法,将混沌复杂动态网络的所有节点镇定到其平衡点.利用李稚普诺夫稳定性理论,得到控制器参数选择条件,以蔡氏混沌电路作为网络节点动态进行仿真研究,证明该方法的有效性.     

Global asymptotic stability for a periodic delay hematopoiesis model with impulses

In this paper, sufficient conditions for the global asymptotic stability of a broad family of periodic impulsive scalar delay differential equations are obtained. These conditions are applied to a periodic hematopoiesis model with multiple time-dependent delays and linear impulses, in order to establish criteria for the global asymptotic stability of a positive periodic solution. The present results are discussed within the context of recent literature. In conclusion, when compared with previous works, not only sharper stability criteria are obtained here, even for models without impulses, but also the usual constraints imposed on the linear impulses are relaxed.     

Asymptotic stability and estimates for the attraction domains of monotone difference systems

We generalize asymptotic stability criteria and estimates for attraction domains in the nonnegative cone for systems of autonomous difference equations with monotone discontinuous right-hand side.     

离散大系统非线性比较方程的稳定性

用矢量李雅普诺夫函数解决大系统的稳定性问题必须要判断矢量比较方程的稳定性.对离散系统,过去只研究过线性驻定比较方程的稳定性.本文全面建立了离散非线性驻定比较方程的各种稳定性判别准则,其中渐近稳定的准则既是充分也是必要的,并由此推得了一个用于C₁类函数的准则,两者均可用来判断离散非线性(驻定或非驻定)系统的非指数稳定以至全局非指数稳定.所有准则均具有简单的代数形式,便于应用.     

Asymptotical stability of partial difference equations with variable coefficients

The dynamical systems considered have scalar state, are multivariate, linear, time-discrete, and time-variable and are described by an initial value problem for a class of evolutionary partial difference equations. The time set is the nonnegative part of the integer lattice in several dimensions. Parts of the asymptotical stability set in the parameter space spanned by the time-variable coefficients are explicitly found. To assess the quality of the sufficient stability criteria, a comparison with the exact stability set is made in an example.     

Exponential stability and estimates for monotone and differential-difference systems

We present necessary and sufficient conditions for the exponential stability in the nonnegative cone and refine exponential estimates for solutions of systems of autonomous difference equations with monotone nondecreasing right-hand sides, including discontinuous ones, as well as for solutions of some class of systems of differential-difference equations with monotonicity. Unlike well-known criteria, the new ones are free of some additional assumptions on the right-hand sides of the considered models other than the original monotonicity conditions. We show that, in the nonsmooth and discontinuous cases, the traditional exponential stability conditions based on ??linearization?? can lead to negative or very coarse results.     

ASYMPTOTIC STABILITY FOR LINEAR AUTONOMOUS DIFFERENTIAL EQUATION WITH SEVERAL DELAYS

Inmathematicalbiology,therehavebeenmanybio1ogicalmodelsinpopu-1ationdynamcis,ecologyandepidemic,whicharedescribedbyautonomousdelaydifferentialequations.Theasymptoticstabilityofthosen1ode1sisinte-resting,whichhavebeenstudiedbymanyauthors[1-4].Thepurposeoft…     

On the behavior of the solutions for linear autonomous neutral delay difference equations

A class of linear autonomous neutral delay difference equations is considered, and some new results on the asymptotic behavior and the stability are given, via a positive root of the correspondng characteristic equation.     

A mathematical analysis of Miller's algorithm

Summary The backward recurrence algorithm of J. C. P. Miller is generalized for homogeneous systems of linear difference equations. Necessary and sufficient conditions for both convergence and stability are given, and the results are demonstrated for scalar equations of arbitrary order. It is shown that the algorithm need not converge to a minimal solution. An application to the calculation of Incomplete Bessel functions is presented.     

The behavior of the solutions of periodic linear neutral delay difference equations

Some new asymptotic, nonoscillation and stability criteria for linear neutral delay difference equations with periodic coefficients and constant delays are given. The results are obtained via a positive root (with suitable properties) of an associated equation which is, in a sense, the corresponding characteristic equation.     

Global stabilization in nonlinear discrete systems with time-delay

A class of scalar nonlinear difference equations with delay is considered. Sufficient conditions for the global asymptotic stability of a unique equilibrium are given. Applications in economics and other fields lead to consideration of associated optimal control problems. An optimal control problem of maximizing a consumption functional is stated. The existence of optimal solutions is established and their stability (the turnpike property) is proved.     

具有可变时滞的非线性非自治差分方程的线性化全局渐近稳定性及其应用

证明了在一定条件下,具有可变时滞的非线性非自治差分方程的全局渐近稳定性可由某种线性差分方程的渐近稳定性确定,给出了这类差分方程全局渐近稳定的充分条件.作为实例,获得了具有可变时滞的离散型非自治广义Log istic方程的全局吸收性判别准则.