The winkling model in the process of drawing was developed on the base of Hills bifurcation-point bulking theory, which became the theoretical basis of the analysis of flange wrinkle and side-wall wrinkle.
本文在Hill分叉点失稳的能量法则的基础上建立了半球形零件拉深成形的皱曲模型，该模型是分析内皱及外皱的理论依据。
The direction and stability of bifurcation are determined by the normal form theory and center manifold theorem. Results of some computer simulations are displayed graphically.
利用中心投影方法我们给出了分岔方向和稳定性的计算公式，并利用计算机对Neimark-Sacker-分岔进行了模拟验证。
Using bifurcation theory, process of effect on the system's stability caused by various uncertain factors is described and analyzed, and some conclusions are drawn as follows. Under the same extent of change of parameters, the effect on stability is greater when cylinder moves forward than that when cylingder moves backward. The longer oil inlet pipe, the worse the stability, and the degree that the stability becomes worse is very small when the length of pipe exceeds a certain value. The bigger the diameter of oil inlet pipe, the better the stability, and the degree that the stability becomes better is very small when the diameter of pipe exceeds a certain value. The stability also becomes bad when the fluid bulk modulus reduces.
利用分岔理论，形象描述和分析了各种不确定因素对系统稳定性的影响过程，得出了一些对系统稳定性设计有用的结论，如：同等程度参数的变化对液压缸正向运动时稳定性的影响比对反向运动时稳定性的影响要大；进油管道越长，系统的稳定性越差，当管道长度超过某一值之后系统稳定性进一步变差的程度非常小；进油管道直径越大，系统稳定性越好，当进油管道直径超过某一值之后系统稳定性进一步变好的程度非常小；液压油弹性模量的降低会使系统稳定性变差。
In this paper transversality idea from singularity theory is applied to discuss the classification of unfoldings of bifurcation problem s.
利用奇点理论中的横切性思想讨论分支问题开折的分类问题，给出了两个开折同构的一个充要条件。
The main contents include:(1) the introduction to the basic concepts of dynamic systems, including attractors, invariant manifold, structural stability, and bifurcations; (2) the main framework of chaotic dyanmics, including the definitions of chaos; (3) the principles of methods of chaos control, including the OGY method, the feedforward method, and kinds of feedback methods in practice; (4) the principle of bifurcation control and methods; (5) chaos synchronization and its application in secure communication; (6) the theory and algorithms of feedback chaotification to generate chaos.
主要教学内容包括：（1）介绍动力系统的基本概念，包括吸引子，不变流形，结构稳定性，分岔等。（2）重点介绍混沌动力学的基本知识，包括混沌的判别方法等。（3）介绍混沌控制的基本原理与方法，包括OGY方法，前馈方法和各种工程反馈控制方法。（4）介绍分岔控制的基本原理与方法。（5）介绍混沌同步及其在通信中的应用。（6）介绍用反馈控制的方法产生混沌的反馈混沌化理论与方法。
Secondly, it interprets the gradual bifurcation cascade sequence of enterprise system evolution, reveals the reason of enterprise system diversity and studies mutation condition during the evolution. The condition is considered to be dominated by internal and external control variable together. And for the first time, the entropy theory and primary factor analysis method are applied in determining the control variable.
其次，阐释了企业系统演化的逐级分岔序列，揭示了企业系统的多样性产生的根源，研究了企业系统演化中的突变条件，认为由企业系统的内外部控制变量共同主导，并首次应用熵理论，加入权重，应用主因子分析的方法对控制参量进行了确定。
In this paper transversality idea from singularity theory is applied to set up the universal unfolding theorem of bifurcation problem s with multiparameter.
利用奇点理论中的横切性思想建立了多参数分支问题的方有开折定理。
The theory of bifurcations of the fixed point was applied to such model, and it was proved that the symmetric periodic motion only has pitchfork bifurcation by the analysis of the symmetry of the Poincaré map.
把映射不动点的分岔理论运用到该模型，并通过分析对称系统的Poincaré映射的对称性，证明了对称型周期运动只能发生音叉分岔。
One of the basic problems in bifurcation theory is to understand the way in which horseshoes created.
如何理解smale马蹄的产生是分枝理论的一个基本问题。
Based on the bifurcation theory of dynamical systems, the existence of their solitary wave, kink and anti-kink wave, and periodic wave solution is proved.
利用平面动力系统分支理论，证明了非线性Schrodinger方程孤立行波、周期波、扭子与反扭子波解的存在性。
Using singularity theory for classifiying smooth germs of function under the action of strong contact equivalence group, problems concerning the existence and number of bifurcation solution of a class of 2nd-order nonlinear system are studied.
借助奇点理论对光滑函数芽在强接触等价群作用下的分类，研究了一类非线性二阶系统边值分支解的存在性和分支解的个数等问题。
On the basis of the bifurcation set of cusp catastrophe theory, the formula to predict the sliding time is derived, which can be applied to forecast landslides. Meanwhile, the formulae of initial sliding velocity are obtained by using energy transformation theory.
根据尖点突变理论的分叉集方程，得到了滑坡突滑时间的计算公式，同时根据能量转化原理，导出了滑体突滑初速度的计算公式。
The analysis for carbon monoxide oxidation reaction via singularity theory and catastrophe theory shows that there are at most three steady states in this system, and the bifurcation diagrams, which describe the relation between state variable y and bifurcation parameter α, can only be monotonic increasing curve in which there is a unique steady state, or S-shaped curve in which there exist multiple steady states.
对一氧化碳和乙烯的同进氧化反应体系，当两个反应均按一级不可逆反应处理时，导出的系统定态模型不能预测体系多定态特性。除一级反应动力学模型外，利用其它可能的动力学模型将无法导出一简洁的系统定态模型方程表达式，这样利用奇异理论和突变理论对系统多定态特性的分析和预测将受到限制。通过对实验结果的模拟，得知
By local bifurcation theory and perturbation theory, the existence of local bifurcating solution and its stability are obtained.
研究了一类带简化的Mound-Haldane反应项的捕食-食铒模型的局部共存解存在性及其稳定性，运用局部分歧理论及特征值扰动理论，得到了该模型的局部分歧解存在的充分条件及其解的稳定性。
A theory on the transition of fixed points and bifurcation points in a SAR system is advanced for the first time.
首次提出了SAR系统中不动点与分叉点的跃迁与漂移的理论。
We exhibit calculating formula about direction of Hopf bifurcation and the bifurcation periodic solutions in the first bifurcated TO and so on with norm form theory and center mainfold theorem, Furthermore, we carry out numerical simulated compution with Mathematica.
我们还用中心流形和规范形理论，给出了在第一个分支点τ_0处的Hopf分支方向，分支周期解的稳定性的计算公式。
Applying the main monomial analysis method and Hopf bifurcation theory to analyzing the equilibrium point of the system, the locally stable or periodically oscillations le adjusting parameters are found.
利用主项分析法和Hopf分支理论时系统的平衡点进行分析，通过调节参数使系统处于局部稳定或者出现周期震荡，给出了能源均衡价格局部稳定性条件和出现Hopf分支的条件，得出能源经济系统的相关结论。
Stability of all equilibrium positions of the front bridge and turning wheel model with three degrees of freedom is studied by the stability theory of ordinary differential equations and numerical analysis method. A Hopf bifurcation does not occar for all equilibriwn positions except the zero position. And changing of some parameters will result in disappearing of the car turning wheel shimmy, and some do not influence the shimmy and some will result in appearing of the second shimmy.
以三自由度前桥转向轮模型为研究对象，利用常微分方程稳定性理论和数值分析得出系统除零平衡点外其他平衡点都不会发生Hopf分岔，而且发现改变某些参数可以完全消除摆振，改变某些参数对系统摆振没有明显影响，改变某些参数还可以引起二次摆振。
ABSTRACT The theory and numerical computation methods of local bifurcation in voltage stability analysis of power system are summarized briefly in this paper.
简要总结了面向电压稳定分析的局部分岔理论及其数值计算方法，初步讨论了应用该法时电力系统的模型问题，介绍了用于计算解支的延拓法，并给出了应用于电力系统分岔计算的延拓法的基本步骤。
The ordinary differential Van der Pol equation and the functional differential Van der Pol equation with delays whose nonlinear terms are in two different forms are compared accordingly in this paper, mainly by means of the planar qualitative theory, the Hopf bifurcation theorem and the cone fixed point theorem in the functional differential equations.
本文对非线性项为两种形式的常微Van der Pol方程和时滞Van der Pol方程分别加以比较，主要利用平面定性理论、Hopf分支理论和锥不动点定理，得到相应方程在周期解和Hopf分支方面上的差异，从而一定程度上体现了常微分方程和泛函微分方程在具体方程上的差异。