Towards a bifurcation theory for nonautonomous difference equations

Although, bifurcation theory of equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. This article contains an approach to overcome this deficit in the context of nonautonomous difference equations. Based on special notions of attractivity and repulsivity, nonautonomous bifurcation phenomena are studied. We obtain generalizations of the well-known one-dimensional transcritical and pitchfork bifurcation.     

Semi-Fredholm problems on periodic solutions of functional-differential equations of neutral type

We consider the problem on periodic solutions for linear systems of functionaldifferential equations of neutral type with periodic coefficients and periodic deviations of the argument. By reduction to associated functional equations, we derive necessary and sufficient conditions under which the problem on periodic solutions for such a system is Fredholm or semi-Fredholm.     

Existence Results for Super-Liouville Equations on the Sphere via Bifurcation Theory

We are concerned with super-Liouville equations on $\mathbb{S}^2$, which have variational structure with a strongly-indefinite functional. We prove the existence of nontrivial solutions by combining the use of Nehari manifolds, balancing conditions and bifurcation theory.     

弹性支承-刚性转子系统同步全周碰摩的分岔响应

基于航空发动机转子系统的结构特点,将航空发动机转子系统简化为一个非线性弹性支承的刚性转子系统．根据Lagrange方程建立了弹性支承-刚性不对称转子系统同步全周碰摩的运动方程;采用平均法进行求解,得到了关于系统振幅的分岔方程;根据两状态变量约束分岔理论,分别给出了系统在无碰摩和碰摩阶段参数平面的转迁集和分岔图,讨论了转子偏心、阻尼对系统分岔行为的影响;应用Liapunov稳定性理论分析了系统碰摩周期解的稳定性和失稳方式,给出了系统参数——转速平面上周期解的稳定范围;该文的研究结果对航空发动机转子系统的设计有一定的理论意义．     

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．     

中立型泛函微分方程的多步法

在探求泛函微分方程的数值解法时,常常设法把常微分方程的许多古典的有效方法经过改造移殖到泛函微分方程。常微分方程的各种数值方法本质上都是力图使变量离散化。所以,Henrici干脆称之为离散变量法。与常微分方程不同,计算泛函微分方程的解在第n点上的值,不仅与前面某k个点上的值有关,而且往往与这些分点之间的值有关。这是推广过程中遇到的一大困难。1964年Feldstein对时滞微分方程提出了所谓连续Euler法,引进了插值思想,使离散方法连续化,克服了上述困难。1973年Castleton和Crimin把这一方法推广到中立型泛函微分方程。特别是Cryer和Tavernini及     

Multiple solutions for boundary value problems of second-order difference equations with resonance

In this paper, we study the existence of multiple solutions for boundary value problems of second-order difference equations with resonance at both infinity and zero by using Morse theory, critical point theory, minimax methods and bifurcation theory.     

Exact multiplicity of solutions to perturbed logistic type equations on a symmetric domain

We apply the imperfect bifurcation theory in Banach spaces to study the exact multiplicity of solutions to a perturbed logistic type equations on a symmetric spatial domain. We obtain the precise bifurcation diagrams.     

系数连续的随机泛函型微分方程解的存在性

本文是［１］的继续，证明了对于随机泛函型微分方程在系数不满足Ｌｉｐｓｃｈｉｔｚ条件时解的存在性。     

A NEW DETECTING METHOD FOR CONDITIONS OF EXISTENCE OF HOPF BIFURCATION

ＡＮＥＷＤＥＴＥＣＴＩＮＧＭＥＴＨＯＤＦＯＲＣＯＮＤＩＴＩＯＮＳＯＦＥＸＩＳＴＥＮＣＥＯＦＨＯＰＦＢＩＦＵＲＣＡＴＩＯＮＳＨＥＮＪＩＡＱＩ（沈家骐）；ＪＩＮＧＺＨＵＪＵＮ（井竹君）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈａｎｄｏｎｇＵｎ...     

具有无界时滞的微分方程解的振动性

具有无界时滞的微分方程解的振动性刘正荣（云南大学数学系和云南省应用数学研究所，昆明６５００９１）俞元洪（中国科学院应用数学研究所，北京１０００８０）基金项目：国家自然科学基金资助项目１９９２年３月２５日收到．一、引言考虑泛函微分方程ｉ＝１其中，．关于...     

一类三维神经元模型的分支研究

考虑一类三维神经元模型的分支问题.利用常微分方程的定性与分支理论的知识,讨论了模型的平衡点个数及其稳定性,主要分析了平衡点的Hopf分支和Bogdanov-Takens分支,并得到了相应的鞍结点分支曲线,Hopf分支曲线与同宿分支曲线.     

Linear stability and Hopf bifurcation in a delayed two-coupled oscillator with excitatory-to-inhibitory connection

We consider the dynamical behavior of a delayed two-coupled oscillator with excitatory-to-inhibitory connection. Some parameter regions are given for linear stability, absolute synchronization, and Hopf bifurcations by using the theory of functional differential equations. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. We also investigate the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. Finally, numerical simulations are given to illustrate the results obtained.     

一类非线性无穷时滞中立型泛函微分方程系统的一致稳定性

本文讨论一类无穷时滞非线性中立型泛函微分方程解的渐近性态与零解的一致稳定性，得到若干简单的稳定性判据。     

一类弹性储液箱同液体耦合晃动问题的分岔行为分析

建立了弹性圆柱型储液箱同液体耦合系统在外激励下的非线性振动方程组．采用多尺度法、奇异性理论研究此非线性振动系统共振解的分岔行为,通过对其分岔行为的分析和讨论,得到了这一系统的多种转迁集和分岔图,建立了系统参数与其拓扑分岔解的联系,并且分析了不同参数下系统的分岔特性,为实现储液器参数的优化控制提供了理论依据．     

Crowding effects on coexistence solutions in the unstirred chemostat

This paper deals with the unstirred chemostat model with crowding effects. The introduction of crowding effects makes the conservation law invalid, and the equations cannot be combined to eliminate one of the variables. Consequently, the usual reduction of the system to a competitive system of one order lower is lost. Thus the system with predation and competition is non-monotone, and the single population model cannot be reduced to a scalar system. First, the uniqueness and asymptotic behaviors of the semi-trivial solutions are established. Second, the existence and structure of coexistence solutions are given by the degree theory and bifurcation theory. It turns out that the positive bifurcation branch connects one semi-trivial solution branch with another. Finally, the stability and asymptotic behaviors of coexistence solutions are discussed in some cases. It is shown that crowding effects are sufficiently effective in the occurrence of coexisting, and overcrowding of a species has an inhibiting effect on itself.     

Generic bifurcation with application to the von Kármán equations

We examine the possible types of generic bifurcation than can occur for a three-parameter family of mappings from a Banach space into itself. Specifically, the general form of the bifurcation equations arising from the von Kármán equations for the buckling of a rectangular plate is investigated. Chow, Hale, and Mallet-Paret (Applications of generic bifurcation. II, Arch. Rational Mech. Anal.67 (1976)) studied the bifurcation of solutions to these equations in a two-parameter setting. These parameters were related to the normal loading and to the compressive thrust applied at the ends of the plate. We introduce a third bifurcation parameter by considering the length of the plate as variable. The generic hypotheses of Chow et al. no longer apply in this three-parameter setting, but modifications and extensions of these hypotheses permit a characterization of the three-parameter bifurcation diagram. The bifurcation sheets of this diagram appear as a natural generalization of the finite collection of arcs comprising the two-parameter diagram. As an example of this theory, an actual three-parameter bifurcation diagram is constructed for a specific form of the von Kármán equations.     

Existence of positive solutions for a prey-predator model with refuge and diffusion

We are concerned with a prey-predator reaction-diffusion model with monotonic functional response and specific refuge size. We discuss both existence and nonexistence of positive solutions of the model by using fixed point index theory and bifurcation theory as the main argument tools. We also discuss bifurcation solutions of which stability is established by using spectrum analysis methods. Moreover, we analyze the effects of refuge size on the dynamics of the model.     

Space–time chaos in a system of reaction–diffusion equations

We find conditions for the bifurcation of periodic spatially homogeneous and spatially inhomogeneous solutions of a three-dimensional system of nonlinear partial differential equations describing a soil aggregate model. We show that the transition to diffusion chaos in this model occurs via a subharmonic cascade of bifurcations of stable limit cycles in accordance with the universal Feigenbaum–Sharkovskii–Magnitskii bifurcation theory.     

一类微生物种群生态数学模型的Hopf分支

讨论了一类具有二阶生长速率的微生物菌群生态数学模型。运用常微分方程空间定性理论的手法,在四维相空间中对该模型进行了深入讨论,判定了平衡点的类型及稳定性,分析了正平衡点的存在及成为O+吸引子的条件。最后讨论了系统小扰动下产生Hopf分支的问题。