平衡点

The conditions for the global asymptotic stability of the equilibrium point are expressed as the stability of the some decomposed matrix.

这种方法以矩阵分解为工具，结合平衡点的渐进稳定判据，用分解矩阵的稳定性决定平衡点的全局渐近稳定性。

If the topology is fixed and time-invariant, the consensus value of discrete-time systems also globally asymptotically converges to the convex combination of initial states, which is identical to continuous systems.

如果多智能体网络系统的拓扑结构没有改变，在离散状态下系统的一致平衡点仍收敛于初始状态的凸组合，并且离散状态下系统的一致平衡点与连续状态下系统的一致平衡点相等。

The first part presents develops of predator-prey systems and gives the model that is investigated in our paper; Chapter 2 presents the results on boundedness of solutions and permanence of system; In chapter 3, we explicitly derive a domain of attraction for the positive equilibrium of system by using Liapunov method; The local stability and the conditions that Hopf bifurcation occurs are obtained in chapter 4; In chapter 5, some explicit fomulae of the direction of Hopf bifurcation and the stability of bifurcating periodic solutions on the center manifold are determined,using normal form and center manifold introduced by Hassard et al; In chapter 6, some numerical simulations are performed to illustrate the analytical results found; Our paper ends with a brief conclusion.

全文共分七章，第1章对捕食-食饵系统的发展背景做了介绍并给出了本文研究的模型；第2章讨论了系统的保持性和解的有界性；第3章研究正平衡点的全局性质，对正平衡点的吸引域进行了精确估计；第4章研究了系统的正平衡点的局部稳定性和Hopf分支存在的条件；第5章利用规范型和中心流形定理得到了一个决定Hopf分支方向、分支周期解的稳定性及其他性质的精确公式；第6章对一个具体模型进行了数值模拟；第7章对全文进行了总结。

It is proved that there is no limit cycle around the positive equilibra by constructing Dulac function, thus global asymptotic stability of the positive equilibria is proved in the first quadrant.

分析了该系统的平衡点性态，利用Dulac函数证明了系统在正平衡点外围不存在极限环，从而证明了正平衡点在第一象限内是全局渐近稳定的。

We discussed the positive invariant set,the exsistence and the stability of the equilibrium by the stability theory of ordinary differential equation and the conditions for the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium are obtained.

本文讨论了模型的正不变集，运用微分方程稳定性理论分析了模型平衡点的存在性及稳定性，得出了疾病消除平衡点和地方病平衡点的全局渐进稳定的充分条件。

In the mean time we also give out a reduction way for nonlinear generalized system F =0 by use of the special projection mappings Q . Based on these we prove that the qualitative structure near the isolated equilibrium point of regular nonlinear generalized system is qualitatively equivalent with the structure near the equilibriun point of the linearization system of nonlinear generalized system, so we give out a way for qualitative analysis near equilibrium point of nonlinear generalized system.

其次，对一般形式的非线性广义系统F=0，利用特殊选择的投影函数Q，给出了一种归约方法，在此基础上，并证明了正则非线性广义系统问题的孤立平衡点的小邻域的定性结构等价于其线性化后的线性广义系统平衡点的定性结构，从而为非线性广义系统的平衡点的定性分析提供了一条途径。

The existence of biological, economic and optimum equilibrium of the system is examined.

证明了系统生态平衡点、生态经济平衡点和最适宜的平衡点的存在性。

In addition, considering the move of equilibrium point caused by parameter perturbation and time delay, the offset of equilibrium point is estimated in order to guarantee that the equilibrium point satisfies the need of design.

另外，考虑参数扰动造成平衡点的摄动，对参数扰动和时滞作用下平衡点的偏移进行了估计，以保证递归神经网络平衡点满足设计要求。

When parameters satisfy some assumptions, Rossler system is a chaotic dynamical system. In order to discuss the dynamical character of Rossler system and use delayed feedback control chaos, in the first, we discuss the stability of the steady states of Rossler system with delayed feedback. When the engenvalues of the linear part are pure imaginary numbers, we obtain the corresponding delay value. The stability of the steady state is lost when the delay passes through the critical value, and simultaneity chaos may vanish, as well as there will be a family periodic solutions bifurcate from the steady states.

在几个参数满足一定的假设条件下，Rossler系统是一个具有混沌特性的动力系统，为了研究该系统的动力学性质并利用时滞实现混沌控制，本文首先对具有时滞的Rossler系统的平衡点进行稳定性分析，当系统的线性部分对应的特征值为纯虚根时，算出相应的时滞τ，得到了平衡点的稳定性会在时滞τ取某些值时发生翻转，同时系统的混沌特性也会出现变化，并且在平衡点处系统经历Hopf分支。

We also conclude that the equilibrium of the prey population will be enlarged due to the Allee effect of the predator, but the Allee effects of the prey may decrease the equilibrium value of the predator, or that of both the predator and prey.

如果捕食者受Allee效应的影响，则猎物种群在平衡点的密度将增大；但是猎物的Allee效应将同时减小捕食者和猎物在平衡点的密度，或者减少捕食者在平衡点的密度。

Breath, muscle contraction of the buttocks; arch body, as far as possible to hold his head, right leg straight towards the ceiling (peg-leg knee in order to avoid muscle tension).

呼气，收缩臀部肌肉；拱起身体，尽量抬起头来，右腿伸直朝向天花板（膝微屈，以避免肌肉紧张）。

The cost of moving grain food products was unchanged from May, but year over year are up 8%.

粮食产品的运输费用与5月份相比没有变化，但却比去年同期高8％。