时滞

Choosing time delay as the control parameter, the stability of trivial equilibrium is discussed by analyzing distribution of the roots of the associated characteristic equation. It is found that Hopf bifurcation occurs from trivial equilibrium when the delay passes through critical values, then the critical values and their relations with system control parameters are obtained.

以时滞量为参数，通过分析对应线性系统特征方程的根的分布，讨论了平凡解的稳定性，得到了平凡解发生Hopf分岔而失稳的临界时滞量以及临界时滞量与系统控制参数的关系。

In this paper, the problem of the robust stability and decay estimations of such control system is considered. Several new criteria are given by applying the differential theorem, the decay rate for this class of system is also obtained.

根据微分方程理论和矩阵指数特性，提出了具有不确定性的时滞控制系统与时滞相关和与时滞无关稳定的充分性判据，并给出系统衰减速率的估计方法。

By transforming the delayed neural model to the describer model and then employing the Lyapunov-Krasovskii stability theorem, linear matrix inequality LMI technique, S procedure, and some algebraic inequality method, a new sufficient condition is derived, which is determined by the coefficients of the model and includes more tuning parameters for determining the globally asymptotical stability of recurrent neural networks with time-varying delay.

首先将要研究的模型转化为描述系统模型，然后利用Lyapunov-Krasovskii稳定性定理、线性矩阵不等式技术、S过程和代数不等式方法，得到了确保时变时滞递归神经网络渐近稳定性的新的充分条件，并将它应用于常时滞神经网络和时滞细胞神经网络模型，分别得到了相应的全局渐近稳定性条件。

A class of affine nonlinear system of single input-single output is investigated. Firstly, the nonlinear system is transformed into an equivalent linear system through a diffeomorphism transformation. The networked control problems for the delays are constant, stochastic and shorter than a sampling period, and long and stochastic are investigated respectively.

针对一类单输入-单输出仿射非线性系统，首先通过微分同胚变换实现了仿射非线性系统的无反馈精确线性化；然后基于精确化线性模型，分别讨论网络延时为常时滞，随机时滞小于一个采样周期和随机长时滞情况的控制问题。

In chapter 4, we investigated the existence and asymptotic behaviors of periodic solutions and steady solutions of a type of reaction diffusion equation under different conditions respectively.

在第四章中，我们分别讨论了一类时滞反应扩散方程在有时滞和无时滞的情况下其平衡态解与周期解的存在性，并得到了解的渐近行为的一些结论。

Those figures indicate that when delay is very small, the stability of steady states doesn't change and chaos still exist. As delay increases, the stability of steady states is changed. When delay is near some critical value, a family of periodic solutions bifurcate from the steady state.

这些系统的相图显示了当时滞为零或者时滞很小时，平衡点的稳定性没有发生变化，于是轨道的混沌震荡也没有发生改变，而当时滞逐渐增大时，系统的混沌特性会发生变化，在τ取某些值时，混沌震荡会演变为一个稳定周期轨道。

Using the theory on delay functional and impulsive differential equation, sufficient condition with time delay for the permanence of the system was obtained. And it was proved, that time delays, pulse vaccination and vertical transmission can bring obvious effects on the dynamics behaviors of the model.

运用脉冲时滞泛函微分方程理论，获得了含有时滞的持久性的充分条件，并且证明了时滞、脉冲免疫与垂直传染对模型的动力学行为能够产生显著的影响。

The paper consists of four chapters:In chaper 1, we introduce the background and signficance, research and actuality on oscillation of functional partial differential equations; we present research subject in this paper;In chaper 2, we discuss oscillatory property of systems of parabolic differential equations with delays and obtain necessary and sufficient conditions for the oscillation of their solutions; we show the difference between oscillatory property of systems of parabolic differential equations with delays and that of systems of partial differential equtions without delays; we explain the main results with examples;In chapter 3, we discuss oscillatory property of systems of functional parabolic differential equations of neutral type; we obtain some sufficient conditions for the oscillation or full oscillation of their solutions under some conditions; we explain the main results with examples;In chapter 4, we discuss oscillatory property of systems of functional hyperbolic differential equations of neutral type; we obtain sufficient conditions for the oscillation or full oscillation of their solutions under some conditions; we explain the main results with examples.

全文共分四章：第一章简要介绍了泛函偏微分方程的振动的背景和意义、对其研究的简单历史和现状，给出了本文的主要研究对象；第二章讨论了一类时滞抛物方程组解的振动性质，获得了判断其所有解振动的一个易于验证的充要条件；指出了这类具有时滞偏差变元的抛物方程组解的振动性质和不具有时滞偏差变元的抛物方程组解的振动性质的差异；并举例对主要结果进行阐明；第三章讨论了一类中立型抛物方程组解的振动性质，获得了在给定的条件下其所有解振动或全振动的若干充分条件；并举例对主要结果进行阐明；第四章讨论了一类中立型双曲方程组解的振动性质，获得了在给定的条件下其所有解振动或全振动的若干充分条件；并举例对主要结果进行阐明。

Specially, in Hilbert spaces, some necessaryand sufficient conditions are given in terms of the uniformly square integrability anduniform boundness of the resolvent on the imaginary axis respectively. Furthermore,we consider this robustness fordifferential equations with unbounded operatorin the delay term.

我们进一步研究了时滞项含无界算子的抽象微分方程的小时滞鲁棒稳定性，获得了一些充分必要条件，并应用所得结论讨论了具解析半群生成元的时滞系统。

It is often used to describe problems in dynamics, control theory, bionomics and economics, epidemiology and many other fields either about its experimental or applied aspects, because it can reflect facts more veraciously using the delay differential equations to solute the above problems considering the influence of time and lag sufficiently.

在时滞微分方程的研究方面，从一般的特征方程到超越特征方程，从无参数的微分方程到有参数的微分方程，从无时滞的微分方程到有一个时滞的微分方程直至两个甚至多个时滞的微分方程为题的研究，经历了一个发展相当快的过程。

But we don't care about Battlegrounds.

但我们并不在乎沙场中的显露。

Ah! don't mention it, the butcher's shop is a horror.

啊！不用提了。提到肉，真是糟透了。