Bogdanov-Takens singularity in the comprehensive national power model with time delays

In this paper, the comprehensive national power model with time delays is studied. The condition that there is only one trivial equilibrium in the model is given. Based on the analysis of the distribution of the eigenvalues at the trivial equilibrium, it is found that the trivial equilibrium is a Bogdanov-Takens singularity. Using the center manifold theory and the normal form method, the normal form with delay and ratio parameters of the model is obtained. Furthermore, the topological structures of the model near the bifurcation point with the variation of these two parameters are given. The associated development situations of the comprehensive national power for some topological structures are discussed. Finally, some numerical simulations are performed to support the analytic results.     

非半简分叉问题的范式

根据文[1]给出了求解非半简分叉问题范式的方法．作为应用实例分析了一般非线性系统的非半简双零特征值问题的范式，给出用原系统系数表达的范式系数．     

退化时滞微分系统的Hopf分支

本文讨论了一般二雏退化时滞微分系统当r≠0时平衡点稳定性的范围，并以滞量r为分支参数研究系统出现Hopf分支的条件．     

赤潮藻类非线性动力学模型的分岔及稳定性研究

选取两种常见赤潮藻类和一种浮游动物,考虑生态环境的富营养化及赤潮藻类与浮游动物的相互作用,建立了多种群赤潮藻类的非线性动力学模型．首次运用现代非线性动力学理论,对模型的稳定性及分岔行为进行了研究．得到了发生Hopf分岔时的分岔参数值,判断了极限环的稳定性,并发现了该模型通过准周期分岔产生混沌．     

一类三维生态动力系统的Hopf分支

考虑一类具偏食习惯的捕食者与被捕食者模型．利用中心流形定理和 Hopf分支理论讨论并证明了该系统在一定条件下产生Hopf分支，得到中心流形、小振幅空间周期解的渐近表达式，同时给出了周期解稳定性判据．     

一类不可压广义neo-Hookean球体的空穴分岔问题的定性研究

研究了一类不可压的广义neo-Hookean材料组成的球体的空穴分岔问题,该类材料可以看作是带有径向摄动的均匀各向同性不可压的neo-Hookean材料,得到了球体内部空穴生成的条件．与均匀各向同性的neo-Hookean球体的情况相比,证明了当摄动参数属于某些区域时,从平凡解局部向左分岔的空穴分岔解上存在一个二次转向分岔点,空穴生成时的临界载荷会比无摄动的材料的临界载荷小．用奇点理论证明了,空穴分岔方程在临界点附近等价于具有单边约束条件的正规形．用最小势能原理分别讨论了空穴分岔解的稳定性和实际稳定的平衡状态．     

离散Leslie—Gower型捕食与被捕食系统的Neimark—Sacker分岔

本文利用映射的分岔理论讨论了离散Leslie—Gower型捕食与被捕食系统的Neimark—Sacker分岔,并通过数值模拟验证了所得结果的正确性。     

Analyses of Bifurcations and Stability in a Predator-prey System with Holling Type-Ⅳ Functional Response

In this paper the dynamical behaviors of a predator-prey system with Holling Type-Ⅳfunctionalresponse are investigated in detail by using the analyses of qualitative method,bifurcation theory,and numericalsimulation.The qualitative analyses and numerical simulation for the model indicate that it has a unique stablelimit cycle.The bifurcation analyses of the system exhibit static and dynamical bifurcations including saddle-node bifurcation,Hopf bifurcation,homoclinic bifurcation and bifurcation of cusp-type with codimension two(ie,the Bogdanov-Takens bifurcation),and we show the existence of codimension three degenerated equilibriumand the existence of homoclinic orbit by using numerical simulation.     

Zero-Hopf Bifurcation at the Origin and Infinity for a Class of Generalized Lorenz System

In this paper, the zero-Hopf bifurcations are studied for a generalized Lorenz system. Firstly, by using the averaging theory and normal form theory, we provide sufficient conditions for the existence of small amplitude periodic solutions that bifurcate from zero-Hopf equilibria under appropriate parameter perturbations. Secondly, based on the Poincar{\''e} compactification, the dynamic behavior of the generalized Lorenz system at infinity is described, and the zero-Hopf bifurcation at infinity is investigated. Additionally, for the above theoretical results, some related illustrations are given by means of the numerical simulation.     

一种改进的范式方法的应用

对ZHANG Wei-yi和K.Huseyin等人提出的改进的范式方法作了进一步发展,引进了不同的分析过程,使其更适合于符合推导语言（如MAPLE）的应用,与过去的方法相比,在实际计算范式时,文中介绍了分析过程更简便,实用,具有更多的优越性,文中3个范例说明了此结论。     

共轭算子法和非线性动力系统的高阶规范形

规范形理论是研究非线性动力系统退化分含的强有力的方法.在本文里我们利用共轭算子法计算了具有幂零线性部分和不具有Z2-对称性的非线性动力系统的2阶、3阶和4阶规范形,讨论了几种余维3退化分含情况下的普适开析问题及其一些全局特性.     

Visualization of Stable Manifolds and Multidimensional Surfaces in the Analysis of Power System Dynamics

Received August 4, 1998; accepted August 7, 1999     

结构种群动力系统

在这篇文章中,我们通过综述近２０年来对生理结构种群动力系统的研究概况,介绍了算子半群,分岔等非线性分析的数学理论,特别是无穷维动力系统理论在种群生态学研究中的应用。我们希望由此引起对如何应用现代数学理论研究种群生态学的更多的思考。     

强Duffing系统的局部分岔分析

本文通过坐标变化和近恒等变化，将强Ｄｕｆｆｉｎｇ方程化成范式，从而可以得到在不同共振条件下的分合方程以及其近似解，应用奇异性理论研究了强Ｄｕｆｆｉｎｇ在开折参数及物理参数平面上的转迁集及其局部分岔图．     

Bifurcation phenomenon of a class of planar codimension 3 polycycleS (2) with two saddles resonating

By means of finitely-smooth normal form theory and the method of infinitesimal analysis, it is proved that the cyclicity of a planar codimension 3 polycycleS(2) is three and the complete bifurcation diagram is provided. Project supported by the National Natural Science Foundation of China and the DEPF of China.     

具有扩散的广义Brusselator系统的Hopf分歧

在齐次Neumann边界条件下,考虑广义Brusselator系统．首先讨论常微分系统Hopf分歧的存在性,得到渐近稳定的周期解．其次讨论具有扩散的偏微分系统,在扩散系数满足一定的条件下,得到超临界的Hopf分歧,并利用规范形理论和中心流形定理给出空间齐次周期解的渐近稳定性．最后,借助Matlab软件进行数值模拟,证明了定理的结论．同时,正平衡态解和空间非齐次周期解的描绘补充了理论分析结果．     

生化反应中一类动力系统的定性分析

继续和完善了文[1-3]的工作,利用常微分方程定性理论,研究了生化反应中的一类动力系统dx并获得了该系统极限环的存在性,不存在性及极限环的存在惟一性的完整结果.     

能源价格系统在随机干扰作用下的分岔研究

采用Khasminskii极限定理,随机平均法和FPK方程,研究了能源价格系统在随机干扰作用下的Hopf分岔特性,得到了分岔参数,并讨论了分岔参数对系统性态的影响.进而得出能源经济系统的相关结论.     

Bifurcation of limit cycles for the quadratic differential system (III)l=n=0

In this paper we will prove that limit cycles for the quadratic differential system (III)l=n=0 in Chinese classification are concentratedly distributed, and that the maximum number of limit cycles around O (0,0) is at least two. This project is supported by the Natural Science Foundation of Educational Committee of Jiangsu Province.