分岔的

7The local and global bifurcations and degenerate bifurcations of codimension threefor a nonlinear oscillator under combined parametric and forcing excitations are studiedin detail.

7研究了参数激励与强迫激励联合作用下非线性振子的局部分岔、余维3退化分岔和全局分岔。

In condensed matter physics,we present an analyticaltopological current theory of point defects which unifies the topologicalquantization,homotopic classification and evolution of defects in oneframework.The number of possible branch lines is shown to be 2s once atmost at the bifurcation point of order parameter with degeneracy s and thetopological charge is conserved during bifurcation processes.

论文在凝聚态物理中给出了缺陷解析的拓扑流理论，将缺陷的拓扑量子化，同伦分类和演化统一在一个框架之内，并证明了在序参量退化度为s的分岔点，缺陷分岔数目最多为2的s次方，而且在分岔过程中拓扑荷是守恒的。

The first and secondary Hopf bifurcation and the chaos motion of the Langford system are controlled and the relationships between bifurcation parameter and control gains are obtained.

本文分别对Langford系统的一次Hopf分岔、二次Hopf分岔及其混沌运动的控制进行了研究，分别得到控制增益与分岔参数之间的解析关系。

According to static stability and avoiding the singularity induced bifurcation, saddle-node bifurcation, and Hopf bifurcation in power systems, this paper presents mathematical models for stability equilibrium solutions based on the semismooth property of spectral functions in mathematics.

针对电力系统中静态稳定和避免奇异诱导分岔、Saddle-Node分岔、Hopf分岔几类典型的系统稳定问题，基于数学上谱函数的半光滑特性，建立了系统稳定平衡解的数学模型。

After undergoing a chaotic region, the period number will add one, in the same time, magnitude of the subharmonic response increases in sequence as the rotational speed ratio increases. When the rotational speed is below the critical speed, the chaotic bands resulted directly from a perioddoubling bifurcation occur among the successive orders of superharmonic responses, then they can return directly to periodic motion by a reversed period-doubling bifurcation.

在转速小于临界转速的情况下，在分岔图中各个连续阶亚谐运动的转换区中分别都出现了经由一个倍周期分岔直接导致的混沌频带，后又直接由一个逆倍周期分岔转化为周期一的运动，同时在三个混沌频带之间都出现了幅值突跳的现象。

By means of singularities theory the transition variety and bifurcation diagram of the bifurcation equation of the system is accuired.

按照奇异性理论方法对分岔方程进行了分析，得到了分岔方程的转迁集和分岔图。

Analysis of power spectrum indicates that the characteristic frequency created by the Hopf bifurcation is equal to the generalized winding number of the critical transverse modes just before the bifurcation.

结果表明，耦合振子的同步混沌态可以发生一种Hopf分岔，在Hopf分岔后，系统的功率谱中出现了一个特征频率，其值恰好等于分岔前临界横模的广义旋转数。

In Chapter three, the direction and stability of the bifurcating periodic solutions are determined by the Nyquist criterion and the graphical Hopf bifurcation theorem.

在第三章，利用奈奎斯特准则和Hopf分岔图示定理分析上述两个模型的Hopf分岔方向与分岔周期解的稳定性。

Stability of solutions of the cavitated bifurcation equation is discussed in each region by using the minimal potential principle, and the catastrophic phenomenon of cavity formation is explained. 2. In Chapter 4, problems of cavitated bifurcation for two spheres, which are composed of two different kinds of imperfect compressible hyper-elastic materials, were examined, respectively. The parametric analytic solutions of the radially deformed function and the parametric cavitated bifurcation solutions are all obtained for the two spheres.

利用奇点理论给出了空穴分岔方程在分岔点的等价正规形，将缺陷参数分成若干个区域，证明了当缺陷参数属于某些区域时，具有缺陷的超弹性材料球体发生空穴分岔时的临界载荷比无缺陷的小；并利用最小势能原理讨论了空穴分岔方程的解在各个参数区域内的稳定性，解释了空穴生成的突变现象。

Also the normal form and thediagram of the Hopf bifurcation are deduced with the normal form direct method.Secondly, the influences of the physical and aerodynamic parameter on the type of theHopf bifurcation are analyzed. Study shows that the "hard" structural non-linearitiesand the longer distance of the suspension point and the center of mass render theairfoil safer, whereas the "soft" ones render the airfoil more unsafe.

同样应用规范型直接法，研究得出系统颤振的Hopf分岔解的规范型以及分岔图；其次，研究了系统结构和气动参数对Hopf分岔类型的影响，结果表明，表明具有&硬&弹簧性质的结构及增大悬挂点与质心的距离都有助于提高安全性，而应该避免应用&软&弹簧性质的结构。

In her eyes, because that is the doorway toher heart * the place where love resides.

女人的漂亮必須從她的眼睛中去看，因為那是她心靈的窗戶和愛居住的地方

I will send some postcards to you.

我会向你发送一些明信片。