Stability analysis of differential equations with maximum

In this paper, we investigate stability of the zero solution of differential equations with maximum by using Lyapunov functions and Razumikhin techniques. Sufficient conditions for stability, uniform stability and asymptotic stability of the zero solution of such equations are found.     

Global Stability of an Autonomous Stochastic Delay Differential Equation with Discontinuous Coefficients

We study the stability and asymptotic stability of the zero solution of autonomous stochastic delay differential equations with discontinuous coefficients by the Lyapunov second method. For the equations under study, we obtain analogs of the Lyapunov stability theorem and the Barbashin–Krasovskii and Zubov asymptotic stability theorems for the zero solution.     

THE STABILITY OF A CLASS OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYS

1 IntroductionConsider the following functional differential equationX'(t) = A(t)x(t) [' C(t,s)x(s)ds, (1)icwhere x E m; A(t) = (ail(t))... is a n x n function matrix, which continuesin [0, co); C(t,s) = (qj(t,8))... is a n x n function matrix, which conti-nues when 0 5 s 5 t < co, and L oo IIC(u,t)lldu continues in [0, co).The problem on the stability for the zero solution of (1) has been studied bymany papers. But in the known results, the boundedness of j: IIC(t, s)lldsor L " IIC…     

Lyapunov's second method in problems of the stability of solutions of systems with impulse effect

A system of ordinary differential equations with impulse effect at fixed moments of time is considered. The system is assumed to have the zero solution. It is shown that the existence of a corresponding Lyapunov function is a necessary and sufficient condition for the uniform asymptotic stability of the zero solution. Restrictions on perturbations of the right-hand sides of differential equations and impulse effects are obtained under which the uniform asymptotic stability of the zero solution of the ‘unperturbed’ system implies the uniform asymptotic stability of the zero solution of the ‘perturbed’ system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability analysis of the zero solution for two class evolution equations

By constructing some suitable Lyapunov-type functionals and applying the theory of the definite-quadratic form, we obtain the stability of the zero solution of two classes of evolution equations with delays, including reaction–diffusion equations and damped wave equations. Our criteria depend on the derivatives of delays. Consequently, when the delays are constants, these criteria are independent of the magnitudes of the delays, so the delays are harmless for the stability of the zero solution.     

On stability and bifurcation of periodic solutions of delay differential equations

The purpose of this paper is to study a class of delay differential equations with two delays. first, we consider the existence of periodic solutions for some delay differential equations. Second, we investigate the local stability of the zero solution of the equation by analyzing the correlocal stability of the zero solution of the equation by analyzing the corresponding characteristic equation of the linearized equation. The exponential stability of a perturbed delay differential system with a bounded lag is studied. Finally, by choosing one of the delays as a bifurcation parameter, we show that the equation exhibits Hopf and saddle-node bifurcations.     

消振器的数学原理

在工程技术中往往采用消振器来消除自激振荡,使设备或机器不受损坏.本文给出了一个消振器的数学模式 我们讨论了如何适当选取方程组(*)的参数c₁,k₁,k₂,使其零解是全局渐近稳定的,得到了方程组(*)的零解全局渐近稳定的若干定理.     

Stability of linear neutral differential equations with delays and impulses established by the fixed points method

In this paper, we consider the impulsive effects on the stability of the zero solution of the linear neutral differential equations with variable delays. By transforming the equations into ones without impulses and using fixed point theory, some sufficient conditions for asymptotic stability and exponential stability of the zero solution are obtained. The paper extends and improves results on sufficient conditions obtained by Jin and Luo (2008) [17], and Ardjouni and Djoudi (2011) [18], which is shown clearly in Example 1. This paper also shows that the impulse intensity and the impulse time both influence the decay rate of the convergence to zero of the solutions. Finally, two examples are given to show applications of some results obtained.     

Development of the Lyapunov function method in the stability problem for functional-differential equations

In the present paper, we consider the stability problem for delay functional-differential equations with finite delay. We suggest a development of the Lyapunov function method involving the use of scalar comparison equations and limit functions and equations. We prove a localization theorem for the positive limit set of a bounded solution and a theorem on the asymptotic stability of the zero solution. We present examples of sufficient conditions for the asymptotic stability of solutions of systems of the first, second, and arbitrary orders.     

On the connected stability of the large-scale systems of dynamical equations on the time scale in critical cases

The connected stability of a large-scale system of dynamical equations on the time scale is investigated. The case where the trivial zero solution of the system is non-asymptotically stable in the linear approximation is considered. The sufficient conditions for the connected asymptotic stability of the zero solution of the system are obtained on the basis of the Lyapunov direct method.     

具有扰动和无界时滞的一维泛函微分方程的渐近稳定性

该文就无界时滞ｒ（ｔ）讨论了带有扰动的一维泛函微分方程的３／２－稳定性,并得到了零解一致稳定和渐近稳定的一些充分性判据．     

一类非线性时滞差分方程的稳定性

得到了一类线性非自治时滞差分方程的零解的一致稳定、一致渐近稳定和全局渐近稳定的充分条件.     

Estimates of solutions to a class of systems of nonlinear delay differential equations

Under study are the systems of nonlinear delay differential equations with periodic coefficients of the linear terms. Some sufficient conditions for the asymptotic stability of the zero solution are established. We obtain the estimates that characterize the decay rate of solutions at infinity and describe the attraction sets of the zero solution.     

Method of limiting equations for the stability analysis of equations with infinite delay in the Carathéodory conditions: II

We consider systems of nonautonomous nonlinear differential equations with the infinite delay. We study the stability properties and the limiting equations whose right-hand sides are defined as the limit points of some sequence in the introduced function space. By using the method of limiting equations, we obtain new sufficient conditions for the asymptotic stability of the zero solution of the considered class of equations.     

Uniform stability at permanently acting disturbances of solutions of Navier-Stokes equations for compressible isothermic fluids

We prove that the uniform stability at permanently acting disturbances of a given solution of the Navier-Stokes equations for viscous compressible isothermic fluid is a consequence of the uniform exponential stability of the zero solution of so-called linearized equations.The research was supported by the grant No. 201/93/2177 of Grant Agency of Czech Republic.     

Stability of solutions to delay differential equations with periodic coefficients of linear terms

Under study are the systems of quasilinear delay differential equations with periodic coefficients of linear terms. We establish sufficient conditions for the asymptotic stability of the zero solution, obtain estimates for solutions which characterize the decay rate at infinity, and find the attractor of the zero solution. Similar results are obtained for systems with parameters.     

LONG-TIME BEHAVIOR OF A CLASS OF REACTION DIFFUSION EQUATIONS WITH TIME DELAYS

The present paper devotes to the long-time behavior of a class of reaction diffusion equations with delays under Dirichlet boundary conditions. The stability and global attractability for the zero solution are provided, and the existence, stability and attractability for the positive stationary solution are also obtained.     

一个新的稳定性判据

本文得到了一个常系数线性差分方程组零解渐近稳定的新判据。     

具有可变脉冲点的脉冲微分方程的稳定性

该文考虑具有可变脉冲点的脉冲微分方程零解的稳定性。通过利用L yapunov函数以及Razumikhin技巧，可以得到关于具有可变脉冲点的脉冲微分方程零 解的一致稳定和一致渐近稳定的充分条件。     

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.