一类具时滞的生理模型的Hopf分支

本文研究了一类简化的具时滞的生理模型的稳定性和Ｈｏｐｆ分支．首先,以滞量为参数,应用Ｃｏｏｋｅ的方法,把Ｒ＾＋分为两个区间,使当滞量属于相应区间时,所考虑的模型的平凡解是稳定或不稳定的,同时得到了Ｈｏｐｆ分支值．然后,应用中心流形和规范型理论,得到了关于确定Ｈｏｐｆ分支方向和分支周期解的稳定性的计算公式．最后,应用Ｍａｔｈｅｍａｔｉｃａ软件进行了数值模拟。     

扰动双中心Hamiltonian系统的分支

本文对一类具双中心的二次Ｈａｍｉｌｔｏｎｉａｎ扰动系统的Ｈｏｐｆ分支、Ｐｏｉｎｃａｒé分支进行了研究，并讨论了能否在双中心同时产生极限环的问题     

Hopf代数的结构定理和对映阶数

本文中，我们把Ｈｏｐｆ代数的结构定理推广到Ｈｏｐｆ代数意义下的同构，从而给出Ｈｏｐｆ代数既约分支的对映阶数，并得到Ｈｏｐｆ代数扩张的对映阶数是任意的．这部分回答了Ｅ．Ｊ．Ｔａｆｔ１９９４年提出的一个问题．     

具时滞的人类呼吸系统模型的稳定性与分支

研究了描述人类呼吸系统的具时滞的二维微分方程的平凡解的稳定性和Hopf分支．利用规范型理论和中心流形定理给出了关于分支周期解的稳定性及Hopf分支方向等的计算公式,且进行了数值模拟计算．     

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．     

具时滞的二维神经网络模型的分支

研究了一类具时滞的二维神经网络模型．通过对该模型的特征方程根的分布分析, 在适当的参数平面上给出了分支图．得到了pitchfork分支曲线是一条直线,进而研究了每个平衡点的稳定性和Hopf分支的存在性．最后,利用规范性方法和中心流形理论,得到了Hopf分支的分支方向和分支周期界的稳定性．     

中立型微分方程零解的稳定性与全局Hopf分支

本文用Ｒｏｕｃｈｅ定理建立起关于一般的超越函数的零点分布定理，以此定理为基础，结合应用吴建宏等用等变拓扑度理论建立起的一般泛函微分方程的Ｈｏｐｆ分支定理，研究了描述无损传输网络线路的中立型微分方程的零解的稳定性和全局Ｈｏｐｆ分支．     

局部对称流形上的数量曲率

本文讨论了无共轭点测地线上的Ｊａｃｏｂｉ声，证明了具非负数量曲率的局部对称的无共轭点流形及具非负数量曲率的具极点的局部对称的流形之数量曲率只能是零。部分解决了Ｅ．Ｈｏｐｆ猜想。     

含分散时滞反馈chen系统的分支分

研究了含分散时滞反馈的Chen系统,利用Routh-Hurwitz准则分析了在弱核及强核情形下平衡点的局部稳定性及Hopf分支的存在性．还运用规范型理论及中心流形定理,得出了包括决定分支周期解的方向、稳定性和周期的清晰的计算公式,其结果可用于混沌控制分析．     

具有捕食者相互残杀项时滞系统的Hopf分支

研究了具有捕食者相互残杀项的时滞系统的Hopf分支,通过选择时滞作为一个分支参数,研究了正平衡点的稳定性和正周期解的Hopf分支.而且通过应用规范型和中心流形的理论,得出了确定分支方向的明确的算法.     

Hopf bifurcation in a size-structured population dynamic model with random growth

This paper is devoted to the study of a size-structured model with Ricker type birth function as well as random fluctuation in the growth process. The complete model takes the form of a reaction-diffusion equation with a nonlinear and nonlocal boundary condition. We study some dynamical properties of the model by using the theory of integrated semigroups. It is shown that Hopf bifurcation occurs at a positive steady state of the model. This problem is new and is related to the center manifold theory developed recently in [P. Magal, S. Ruan, Center manifold theorem for semilinear equations with non-dense domain and applications to Hopf bifurcation in age-structured models, Mem. Amer. Math. Soc., in press] for semilinear equation with non-densely defined operators.     

具有三重零奇异的时滞微分方程的分支

主要研究三重零奇异的判定和在R~n上零特征根对应的广义特征空间,利用中心流形简化和规范型计算得到参数时滞微分方程的简化形式,对应于文[A note on the triple zero linear degeneracy:Normal forms,dynamical and bifurcation behaviour of an unfolding.Int J Bifur and Chaos,2002,12:2799-2820]中的结果具体分析具有三重零奇异的参数时滞微分方程的分支行为,并给出一例子来阐述得到的结果.     

具有三重零奇异的时滞微分方程的分支

主要研究三重零奇异的判定和在Rn上零特征根对应的广义特征空间,利用中心流形简化和规范型计算得到参数时滞微分方程的简化形式,对应于文[A note on the triple zero linear degeneracy:Normal forms,dynamical and bifurcation behaviour of an unfolding.Int J Bifur and Chaos,2002,12:2799-2820]中的结果具体分析具有三重零奇异的参数时滞微分方程的分支行为,并给出一例子来阐述得到的结果.     

Stability and global Hopf bifurcation in a delayed food web consisting of a prey and two predators

This paper is concerned with a predator–prey system with Holling II functional response and hunting delay and gestation. By regarding the sum of delays as the bifurcation parameter, the local stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. We obtained explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation. Using a global Hopf bifurcation result of Wu [Wu JH. Symmetric functional differential equations and neural networks with memory, Trans Amer Math Soc 1998;350:4799–4838] for functional differential equations, we may show the global existence of the periodic solutions. Finally, several numerical simulations illustrating the theoretical analysis are also given.     

一类离散生态经济系统稳定性与分支研究

本文研究了一个离散生态经济模型的稳定性和分支问题.利用离散奇异系统理论,中心流形定理及Neimark-Sacker分支理论,得到了系统关于不动点的稳定性和Neimark-Sacker分支的有关结果,并与相应的连续模型进行对比分析.推广了文献[5]的结果.     

Hopf bifurcation direction of a delayed bacteria-immunity system with quorum sensing mechanism

Considering the mechanism of quorum sensing, we formulate a bacteria-immunity model to describe the competition between bacteria and immune cells on the basis of Zhang’s model (see [9] for more details). A time delay is introduced to characterize the time in which bacteria receive signal molecules and then combat with immune cells. In the sequel, the length of delay which preserves the stability of the positive equilibrium is estimated, and the existence of Hopf bifurcation when the delay crosses through a critical value is investigated. Further, by using the normal form theory and center manifold theory, the explicit formulae are calculated which determine the stability, the direction and the period of bifurcating periodical solutions. Finally, numerical simulations are employed to verify the mathematical conclusions.     

Hopf bifurcation analysis of a food-limited population model with delay

The dynamics of a food-limited population model with delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived, using the theory of normal form and center manifold. Global existence of periodic solutions is established by using a global Hopf bifurcation result due to [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799–4838].     

Triple-zero bifurcation in van der Pol’s oscillator with delayed feedback

In this paper, we study a classical van der Pol’s equation with delayed feedback. Triple-zero bifurcation is investigated by using center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm forms at the triple-zero bifurcation and show that the model can exhibit transcritical bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and zero-Hopf bifurcation. Some numerical simulations are given to support the analytic results.     

Brusselator模型的扩散引起不稳定性和Hopf分支

研究了Brusselator常微分系统和相应的偏微分系统的Hopf分支,并用规范形理论和中心流形定理讨论了当空间的维数为1时Hopf分支解的稳定性．证明了:当参数满足某些条件时,Brusselator常微分系统的平衡解和周期解是渐近稳定的,而相应的偏微分系统的空间齐次平衡解和空间齐次周期解是不稳定的;如果适当选取参数,那么Brusselator常微分系统不出现Hopf分支,但偏微分系统出现Hopf分支,这表明,扩散可以导致Hopf分支．     

Simple bifurcation of coupled advertising oscillators with delay

In this work, the singular bifurcation of a ring of three coupled advertising oscillators with delay, each of them being an advertising model, is considered. The center manifold reduction and normal form method are employed to study the bifurcation from the double-zero singularity which is induced by the coupled strength. Numerical simulation supports the analysis results.