查询词典 bifurcation point

In this thesis, we take the delay r as a bifurcation parameter to investigate the Hopf bifurcation phenomenon in system (1). By analyzing the associated characteristic transcendental equation of system (2), and using the Hopf bifurcation theorem, we obtain one condition for the existence of Hopf bifurcation in system (1). Furthermore, based on the center manifold theorem and the method of normal form, some interesting results about the properties of Hopf bifurcation are obtained, including the direction of Hopf bifurcation and stability of Hopf bifurcating periodic solutions.

我们以时滞T作为分支参数，研究了系统(1)的Hopf分支现象：通过分析系统(2)的特征超越方程，结合利用Hopf分支定理获得了系统(1)的Hopf分支存在的一个条件；利用中心流形定理和正规形方法分析了系统(1)的Hopf分支的性质，包括分支的方向和分支周期解的稳定性。

In chapter three, the new concepts of the bifurcation point and asymptotic bifurcation point of compact continuous operators in Menger PN-spaces are introduced. Some sufficient conditions for the existence of the bifurcation point and asymptotic bifurcation point are obtained. And the problems on the intrinsic value and intrinsic element of compact continuous operators are studied in Menger PN-spaces. A series of theorems on the existence of the intrinsic value and intrinsic element are obtained.

第三章，在概率赋范线性空间中提出非线性算子的歧点和渐近歧点的新概念，获得了非线性算子存在歧点和渐近歧点的充分条件；讨论了Menger概率赋范线性空间中非线性算子的固有值和固有元问题，得到了非线性算子存在固有值和固有元的一系列充分条件。

Algorithm application of neural networks.Ⅲ Implementation of neuro-computers. The main contribution of the dissertation can be summarized as follows: 1 Hopf bifurcation of three kind of neural networks are discussed in detail, including type of discrete time delay, type of time delay with weak kernel and strong kernel as well as the proof of existence of bifurcation. Other problems such as asymptotic stability of bifurcation periodic solution, algorithm of determining the bifurcation direction, asymptotic stability and style of periodic solution are also studied. The average time delay is chosen as the bifurcation parameter, phenomena pertinent to system states of the continuous time delay network with strong kernel evolving from stable to oscillating, then back to stable again are observed.

论文的主要创新之处可以归纳如下： 1）针对目前国内外对神经网络的分岔研究较少的情况，论文中详细讨论了带离散时延神经网络、带弱核的连续时延神经网络、带连续分布时延且具有强核的神经网络的Hopf分岔现象，从理论上证明了Hopf分岔的存在性，并研究了分岔周期解的渐近稳定性，得到了确定周期解的渐近稳定性、分岔方向、周期解的渐近形式的算法；用平均时延作为分岔参数，发现带强核的连续时延神经网络中存在着系统的状态由稳定变化到振荡现象，当继续增加平均时延参数时，又从振荡变为稳定这一特殊的动力学现象。

What differ from that of the researchers in the past is the consideration of the direction of Hopf bifurcation with Dirichlet boundary condition , that is, the conditions when the Hopf bifurcation periodic solutions are orbitally asymptotically stable with asymptotic phase.Our basic idea come from the results given by Tang[30]. We adopt the method used by Hassard in the Hopf bifurcation theorem in [24], which is to judge the existence and the direction of Hopf bifurcation, and the stability of the Hopf bifurcation periodic solutions.

与以前的研究者不同的是，本文不仅讨论了在Dirichlet边界条件下系统的Hopf分支在以扩散系数为分支参数的情况下的存在性和分支周期解的空间非齐次性，而且更多的关注了系统在Dirichlet边界条件下Hopf分支解的稳定性及Hopf分支的方向，也即系统定态解在何种条件下，当分支参数如何变化时产生Hopf分支且分支周期解是带渐近位相轨道渐近稳定的。

We give the detail analysis and show that the system exhibits subcritical pitchfork bifurcation, supercritical pitchfork bifurcation and transcritical bifurcation in the different regimes of control parameters,we also give the condition of the occurrence of bifurcation from the trivial solution,and consider the bifurcation directions.

其次，以前的研究工作主要是针对反应项是纯粹的二次项(p=0)或纯粹的三次项(p=1)的反应扩散系统。本文中取p∈[0，1]为参数，得到了比Auchmuty等人的工作更丰富的结论。

The existence of saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation is studied clearly, and high codimension bifurcation may emerge for some special parameters.

文中我们全面分析了鞍结分支、Hopf分支和Bogdanov-Takens分支在各种参数条件下的存在性，还发现在一定参数条件下可能出现高余维分支。

RESULTS: The imaging features of CBT were masses located at the bifurcation of the carotid artery, with abundant blood supply in the tumor, the internal and external carotid arteries shifted, the bifurcation of the carotid artery increased like goldcup in DSA. The imaging features of the neurilemmoma were masses located behind the bifurcation of the carotid artery, and the carotid bifurcation was normal, no blood supply in the tumor, but carotid arteries shifted.

结果：颈动脉体瘤的CDFI和DSA特征为：颈总动脉分权处单发性肿块，并与其紧密相连，瘤体内血流极为丰富，瘤体使颈外动脉向前内移位、颈内动脉向后外移位，颈内、外动脉分权角度增大明显，DSA呈高脚杯状改变；神经鞘瘤紧邻颈动脉，颈总动脉分权处角度无改变，瘤体内血流信号不丰富，瘤体使颈动脉向前、外移位。

There are a lot of male friends, the experience of urine bifurcation , from a clinical point of view, for the bifurcation of urine is, not all patients with urinary bifurcation is caused due to disease, because there are some other reasons.

很多男性朋友都有出现尿分岔的经历，从临床角度来看，对于尿分岔来说，不一定所有尿分岔患者都是由于疾病引起的，因为也有一些其它原因造成的。

The main advantages of the methodare that(1)this method is valid for all kinds of bifurcation pointswith the codimension ≤3;(2)it does not use any informations forsecond order derivatives and does not need the bifurcation point to beknown accurately in implementation;(3)it is sufficient to implementthe method that a standard continuation procedure is only used;and(4)derivation of the method is independent on the asymptotic expan-sions of the bifurcation solutions.

这种方法的主要优点是：(1)适应于一切余维数不超过3的分支点；(2)不需要任何二阶导数的信息，也不必精确地知道分支点的位置；(3)只需使用标准延拓软件就可以实现；(4)方法不依赖于分支解的渐近展式。

The cavitated bifurcation equation that describes cavity formation and growth in the interior of the sphere is obtained. It is proved that there exists a unique bifurcation point on the trivial solution of the cavitated bifurcation equation.

本文利用分岔理论和动力学理论中的基本观点、方法和结论，对具有缺陷的超弹性材科组成的球体内部的空穴生成和增长的拟静态问题以及空穴生成后随时间的运动规律等方面进行了系统的研究，得到了一些新的结论，同时给出了相应的数值分析。

Now she was hungry and angry.She began to smoulder.

现在她又饿又气，她开始流露难以抑制的怒火。

You have placed our iniquities before You, Our secret sins in the light of Your presence.

诗90：8 你将我们的罪孽摆在你面前、将我们的隐恶摆在你面光之中。