Asymptotic stability properties of θ-methods for delay differential equations

Deals with the asymptotic stability properties of θ methods for the pantograph equation and the linear delay differential algebraic equation with emphasis on the linear θ methods with variable stepsize schemes for the pantograph equation, proves that asymptotic stability is obtained if and only if θ>1/2, and studies further the one leg θ method for the linear delay differential algebraic equation and establishes the sufficient asymptotic ally differential algebraic stable condition θ=1.

On the Stability and Boundedness of Solutions of Certain Non-Autonomous Delay Differential Equation of Third Order

In this paper, we study certain non-autonomous third order delay differential equations with continuous deviating argument and established sufficient conditions for the stability and boundedness of solutions of the equations. The conditions stated complement previously known results. Example is also given to illustrate the correctness and significance of the result obtained.

GLOBAL ASYMPTOTIC STABILITY FOR A NONLINEAR DELAY DIFFERENCE EQUATION

In this paper,a sufficient condition for the global asymptotic stability of the solutions of the following nonlinear delay difference equation is obtained, xn+ 1=xn+xn- 1xn- 2 +a xnxn- 1+xn- 2 +a, n =0 ,1 ,..., where a∈ [0 ,∞ ) and the initial values x- 2 ,x- 1,x0 ∈ (0 ,∞ ) .As a special case,a conjecture by Ladas is confirmed.

ON STABILITY OF SOLUTIONS OF CERTAIN FOURTH-ORDER DELAY DIFFERENTIAL EQUATIONS

By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established result is less restrictive than those reported in the literature.

Stability analysis of a noise control system in a duct by using delay differential equation

The paper deals with the criteria for the closed- loop stability of a noise control system in a duct. To study the stability of the system, the model of delay differential equation is derived from the propagation of acoustic wave governed by a partial differential equation of hyperbolic type. Then, a simple feedback controller is designed, and its closed- loop stability is analyzed on the basis of the derived model of delay differential equation. The obtained criteria reveal the influence of the controller gain and the positions of a sensor and an actuator on the closed-loop stability. Finally, numerical simulations are presented to support the theoretical results.

THE NUMERICAL STABILITY OF THE BLOCK θ-METHODS FOR DELAY DIFFERENTIAL EQUATIONS

This paper focuses on the numerical stability of the block θ methods adapted to differential equations with a delay argument. For the block θ methods, an interpolation procedure is introduced which leads to the numerical processes that satisfy an important asymptotic stability condition related to the class of test problems y′(t)=ay(t)+by(t-τ) with a,b∈C, Re(a)<-|b| and τ>0. We prove that the block θ method is GP stable if and only if the method is A stable for ordinary differential equations. Furthermore, it is proved that the P and GP stability are equivalent for the block θ method.

STABILITY CRITERIA IN TERMS OF TWO MEASURES FOR DELAY DIFFERENTIAL EQUATIONS

By using the variational Liapunov method, stability properties in terms of two measures for delay differential equations are discussed. In the case that the unperturbed systems are ordinary differential systems, employing auxiliary measure h*(t, x), criteria on nonuniform and uniform stability in terms of two measures for delay differential equations are established.

Fixed Points and Asymptotic Properties of Neutral Stochastic Delay Differential Equations

This paper discusses a linear neutral stochastic differential equation with variable delays. By using fixed point theory, the necessary and sufficient conditions are given to ensure that the trivial solution to such an equation is pth moment asymptotically stable. These conditions do not require the boundedness of delays, nor derivation of delays. An example was also given for illustration.

Numerical stability of solution to delay differential equations under resolvent condition for Runge-Kutta methods

Deals with the application of Kreiss resolvent condition in the error growth analysis of numerical methods, and studies the stability of Runge Kutta method in respect of Kreiss resolvent condition with emphasis on the study on the subclass of collocation methods with abscissas in [0,1] by applying the methods to the test equation U′(t)=λU(t)+μU(t-τ)τ>0 with complex constraints μ and λ, and proves under some assumptions on the R K methods that the error growth is uniformly bounded in the stability region.

THE STABILITY OF LINEAR MULTISTEP METHODS FOR SYSTEMS OF DELAY DIFFERENTIAL EQUATIONS

This paper deals with the numerical solution of initial value problems for systems of differential equations with a delay argument. The numerical stability of a linear multistep method is investigated by analysing the solution of the lest equation y’(t)=Ay(t) + By(1-t),where A,B denote constant complex N×N-matrices,and t】0.We investigate carefully the characterization of the stability region.

NGP_G-STABILITY OF LINEAR MULTISTEP METHODS FOR SYSTEMS OF GENERALIZED NEUTRAL DELAY DIFFERENTIAL EQUATIONS

The stability analysis of linear multistep methods for the numerical solutions of the systems of generalized neutral delay differential equations is discussed. The stability behaviour of linear multistep methods was analysed for the solution of the generalized system of linear neutral test equations, After the establishment of a sufficient condition for asymptotic stability of the solutions of the generalized system, it is shown that a linear multistep method is NGP(G)-stable if and only if it is A-stable.

THE STABILITY OF θ-METHODS FOR PANTOGRAPH DELAY DIFFERENTIAL EQUATIONS

This paper deals with the numerical solution of initial value problems for pantograph differential equations with variable delays. We investigate the stability of one leg θ-methods in the numerical solution of these problems. Sufficient conditions for the asymptotic stability of θ-methods are given by Fourier analysis and Ergodic theory.

Globally Asymptotic Stability of Equilibrium of a Dynamical Model with Infinite Delay

By using a criterion for asymptotic stability in Banach space BC, a group of sufficient condi-tioas for a dynamical model with infinite delay which is derived from hematology were obtained, which refined the result in the reference [ 10] got by the second author herself.

A semigroup approach for numerical solution of delay differential equations

Investigates the asymptotic stability of numerical solution of linear delay differential equations(DDEs) of the form y′(t)=Ly(t)+My(t-τ), t>0, y(t)=(t), -τ≤t≤0, where τ>0, L, M∈ C n×n  and ∈ (,C n), which is formulated using the semigroup instead of classical step by step methods for numerical solution of this equation, as an abstract Cauchy problems and then discretized in a systems of ordinary differential equations(ODEs), and examines the asymptotic stability properties with respect to the class of DDEs with the complex coefficients which preserve the stability properties for a fixed delay.

UNIFORM ASYMPTOTIC STABILITY OF SOLUTIONS TO NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper, we consider neutral functional differential equations with infinite delay. Sufficient and necessary criteria for the g-uniform asymptotic stability of solutions to the system in the phase space (Cg,|·|g) are established.

Asymptotic Stability in the Large of Zero Solution of Second Order Nonlinear Differential Equation

There are many works on the asymptotic stability of second dimensional nonlinear differential equation. In particular, these results only concern with the system which includes one or two terms, whereas few works concern with system which includes more than two terms. In this paper, system which includes four nonlinear terms are studies. We obtain the global asymptotic stability of zero solution, and discard the condition which require the Liapunov function trends to infinity, and only require that the positive orbit is bounded.

ASYMPTOTIC STABILITY FOR GAUSS METHODS FOR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

In [4] we proved that all Gauss methods areNtau(0)-compatible for neutral delay differential equations (NDDEs) of the form y'(t) = ay(t) + by(t-tau) + cy'(t-tau), t >0, (0.1) y(t) = g(t), -tau less than or equal to t less than or equal to 0, where a, b, c are real, tau > 0, g(t) is a continuous real valued function. In this paper we are going to use the theory of order stars to characterize the asymptotic stability properties of Gauss methods for NDDEs. And then proved that all Gauss methods are Ntau(0)-stable.

Asymptotic stability for nonautonomous delay differential equations

MANY dynamic population models can be transformed to the form of delay differential equationx（t） + λx（t） + f（t,x（t-τ1）,….,x（t-τm））=0,t≥0,（1）where the biologically meaningful equilibrium of the original equation is transformed into thezero equilibriurn of （1）. Throughout this note, we assume that λ】0,τi】0（i=1,…,m ）,

Extreme Stability of Functional Differential Equations with Finite Delay

This paper obtained some theorems that can ascertain the zero solution of functional differential equations are extremely uniformly stable, extremely asymptotically stable or extremely uniformly asymptotically stable. In the obtained theorems, the derivative of Liapunov function on t along the solutions of functional differential equations is not required to be always negative, especially, it may be even positive.

Criteria for asymptotical stability independent of delay differential equation

Asymptotical stability independent of delay differential equation can be expressed in terms of zero criteria for polynomials in two independent complex variables. The necessary and sufficient conditions are given which differ from those obtained in the former literature.