分岔

In the first part,we use adjoint operator method and matrixrepresentation method to compute normal forms of higher order for nonlinear dynamicalsystems with Z2—symmetry and Z2—asymmetry.

在第二部分里，利用多尺度法、规范形理论、普适开折理论、Melnikov方法和Petrov方法分别研究了参数激励作用下非线性振子、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2退化分岔、余维3退化分岔和全局分岔。

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。

Moreover, in the parameter plane determined by some two parameters,we discuss codimension two bifurcations caused by the intersections of two bifurcating curves.

同时，在两个参数确定的半参数平面上，我们通过分岔(来源：46e646ABC论文网www.abclunwen.com)曲线的交点讨论了余维2分岔的存在性，即等变Pitchfork-Hopf分岔和Hopf-Hopf分岔。

We use the method of multiple scales to study local bifurcations of this system and obtainbifurcating response equation,transition sets and eight different diagrams of bifurcatingresponse curves.We find that some singular points in bifurcation set are codimension twodegenerate.

首先利用多尺度法研究了这个系统的局部分岔问题，得到了分岔响应方程、局部分岔集和8种分岔响应曲线，并且发现在分岔集上有一些奇点是余维2退化的。

The grazing bifurcation point is found in terms of the conditions for the existence of grazing orbit.The bifurcation diagrams are gained through previous piecewise smooth map and numerical calculation,they are similar structurally.

利用擦边分岔条件找到了此系统的擦边分岔点，利用推导的擦边分岔的分段映射和直接数值仿真获得不连续分岔图，两者得到的分岔图在结构上是类似的。

A class of fast-scale bifurcations appearing around the maximum value of the input voltage is discovered for the first time by experiment. Then the influence of the fast-scale bifurcations on the performance of the PFC converters is discussed. Whether the fast-scale bifurcations will appear and how wide the bifurcation region depend on the characteristics of the current loop output. Therefore, the main factors that affect the current loop output are analyzed, including the current loop parameters, inductance, and root-mean-square value of the input voltage.

通过实验，该文首次发现了一类在输入电压最大值附近发生的快时标分岔；详细讨论了快时标分岔对PFC变换器性能的影响；由于电流环输出的特点决定了变换器是否会出现快时标分岔以及分岔区域的大小，该文分析了影响电流环输出的主要因素，包括电流环参数、电感值和输入电压有效值。

Melnikov method is employed to study the bifurcations and chaos of homoclinic and subharmonic orbits for two pendulum systems.

用Melnikov方法研究具有弱阻尼与参数激励的单摆以及倒摆运动的同宿轨道分岔、次谐分岔和混沌现象，得到了发生同宿轨道分岔、次谐分岔和混沌的临界参数，并将所得理论结果与倒摆运动的实验结果进行了比较与讨论。

The qualitative analysis and numerical results show that a positive period-1 solution bifurcates from the semi-trivial periodic solution through a fold bifurcation, a positive period-2 so...

定性分析和数学模拟表明，半平凡周期解通过fold分岔分岔出正周期1解，正周期1解通过flip分岔分岔出正周期2解，再通过一系列flip分岔通向混沌。

The characteristics of Hopf bifurcation from the origin, including the existence condition, the direction as well as the stability of bifurcating periodic solutions are discussed in detail, which can be demonstrated by the numerical simulations.

讨论了快慢两时间尺度下超混沌Lorenz系统原点的稳定性问题，分析了原点的Hopf分岔，包括Hopf分岔的存在性，分岔方向以及分岔周期解的稳定性等问题，并用数值例子对所得到的结果加以验证。

In Chapter 3, problems of cavitated bifurcation for a solid sphere composed of imperfect incompressible hyper-elastic materials, subjected to a surface tensile dead load, are examined.

证明了空穴分岔方程的平凡解支上存在唯一的分岔点；给出了左分岔和右分岔的判别条件，并指出了向左分岔的空穴分岔解还会发生二次转向点分岔。

The concept of equivalent rotationally rigidity is offered and the formula of rotationally rigidity is obtained.

主要做了如下几个方面的工作：对伸臂位于顶部的单层框架—筒体模型进行分析，提出了等效转动约束的概念和转动约束刚度的表达式。

Male cats normally do not need aftercare with the exception of the night after the anesthetic.

男猫通常不需要善后除了晚上的麻醉。