关于一类二次微分系统的概周期解及周期解（英文）

周期解及概周期解问题在生物科学和其他领域具有重要价值.探究一类二次微分系统的周期解及概周期解,分析了附有周期系数系统的周期解的存在性,证明了附有概周期系数系统的概周期解的存在唯一性及全局吸引性.     

一类中心型半连续动力系统的周期解

讨论了一类中心型半连续动力系统的阶1周期解和阶2周期解的存在性、个数及其稳定性,给出该中心型系统存在唯一、两个、无穷多个阶1周期解和阶2周期解的条件,并给出了相应理论结果的数值模拟.     

食饵具有周期强迫的捕食模型

建立并研究了一类具有周期强迫和脉冲扰动的捕食模型,通过理论分析和数值模拟,得到了食饵灭绝周期解全局渐近稳定和系统持久的充分条件,利用分支理论证明了边界周期解附近会分支出正周期解.     

具周期脉冲的周期差分系统的全局性态

本文建立了具有常数脉冲和周期脉冲的周期差分系统,得到了常数脉冲系统全局稳定周期解存在的充分条件,并证明了周期脉冲的周期系统的周期解是全局吸引的。     

具有周期传染率的SIR传染病模型的周期解

考虑了具有周期传染率的SIR流行病模型,定义了基本再生数^-R0=β／（μ＋γ）,分析了该模型的动力学性态,证明了当^-R0〈1时无病平衡点是全局稳定的;^-R0〉1时,无病平衡点是不稳定的,模型至少存在一个周期解。对小振幅的周期传染率模型,给出了模型周期解的近似表达式,证明了该周期解的稳定性,最后做了数值模拟,结果显示周期解可能是全局稳定的。     

一类捕食-被捕食系统周期解的存在性

对于二维非自治捕食-被捕食系统,本文讨论了它的解的有界性,进而得到周期解和正周期解的存在性。     

含有两个时滞项的退化时滞微分方程周期解

讨论了含有两个时滞项退化时滞微分方程的周期解的问题,特别的,给出了此类方程存在非常数周期解的充要条件,并对二维退化微分方程给出了非常数周期解存在性的代数判据,并在最后给出一个例子验证了判据的有效性.     

时标上具有HollingⅢ型功能反应的捕食模型的周期解

研究了时标上一类具有Holling型功能反应的捕食模型.运用时标上连续拓扑度定理,得到了系统存在周期解的充分条件,从而使系统的连续时间情形和离散时间情形的周期解问题得到了统一,该方法可广泛应用于研究微分方程和差分方程的周期解的存在问题.     

高维非自治系统的周期解

本文通过建立并利用齐次线性方程解的估计公式,获得了周期系统（1）的周期解的存在性、唯一性定理,对周期系统（2）给出了一个平稳振荡定理,最后给出了实例。     

双营养Chemostat模型周期解的全局吸引性

研究含有时滞的双营养单种群Ｃｈｅｍｏｓｔａｔ模型周期解的全局吸引性,首先利用强正、凹算子理论给出了系统存在唯一正周期解的充分条件,然后利用泛函微分方程的单调理论得到了正周期解的全局吸引性。     

时标上一捕食二食饵系统的周期解（英文）

研究时标上一捕食二食饵系统.运用时标上Gaines和Mawhin的连续拓扑度定理,得到了系统存在周期解的新的充分条件.其研究方法可以广泛地运用来研究微分或者差分方程的周期解存在性问题.     

几类具有无穷时滞泛函微分方程正周期解的存在性

运用Leray-Schauder不动点定理研究具有无穷时滞的泛函微分方程的正周期解的存在性问题,获得了存在正周期解的充分条件,改进了文献[3]中的结果.     

固定周期脉冲微分方程到状态依赖脉冲的转化及应用

本文研究了一类二维状态依赖脉冲微分方程的阶1周期解存在性和轨道稳定性条件．然后,将一维固定周期脉冲的微分方程转化为二维状态依赖脉冲微分方程,研究其阶一周期解的存在性和稳定性．作为应用,我们研究了固定周期常数收获的Logistic方程的动力学性质,以及两个固定周期注射药物单室扩散模型的动力学性质．     

The geometrical analysis of a predator-prey model with two state impulses

Using successor functions and Poincaré-Bendixson theorem of impulsive differential equations, the existence of periodical solutions to a predator-prey model with two state impulses is investigated. By stability theorem of periodic solution to impulsive differential equations, the stability conditions of periodic solutions to the system are given. Some simulations are exerted to prove the results.     

Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations II. Singularity theoretic approach and highly degenerate bifurcations

In the Hodgkin-Huxley equations (HH), we have identified the parameter regions in which either two stable periodic solutions with different amplitudes and periods and an equilibrium point or two stable periodic solutions coexist. The global structure of bifurcations in the multiple-parameter space in the HH suggested that the bistabilities of the periodic solutions are associated with the degenerate Hopf bifurcation points by which several qualitatively different behaviors are organized. In this paper, we clarify this by analyzing the details of the degenerate Hopf bifurcations using the singularity theory approach which deals with local bifurcations near a highly degenerate fixed point. Received: 23 April 1999 / Accepted in revised form: 24 September 1999     

Mathematical aspects of peridic catatonic schizophrenia

The mathematical study of periodic catatonic schizophrenia initiated by Danziger and Elmergreen is generalized by considering the class of differential equations that could be used to describe the periodic behavioral symptoms and periodic variation of biochemical levels in periodic catatonic schizophrenia. The existence of asymptotically stable periodic solutions is studied mathematically and the physical significance of such periodic solutions is discussed. The occurrence of relaxation oscillations is briefly considered. The research in this paper was supported partly by the U.S. Army Research Office (Durham) (Grant No. DA-ARO-D-31-124-72-C69) and partly by a Rutgers Research Council Faculty Fellowship.     

Dynamic reduction with applications to mathematical biology and other areas

In a difference or differential equation one is usually interested in finding solutions having certain properties, either intrinsic properties (e.g. bounded, periodic, almost periodic) or extrinsic properties (e.g. stable, asymptotically stable, globally asymptotically stable). In certain instances it may happen that the dependence of these equations on the state variable is such that one may (1) alter that dependency by replacing part of the state variable by a function from a class having some of the above properties and (2) solve the 'reduced' equation for a solution having the remaining properties and lying in the same class. This then sets up a mapping Τ of the class into itself, thus reducing the original problem to one of finding a fixed point of the mapping. The procedure is applied to obtain a globally asymptotically stable periodic solution for a system of difference equations modeling the interaction of wild and genetically altered mosquitoes in an environment yielding periodic parameters. It is also shown that certain coupled periodic systems of difference equations may be completely decoupled so that the mapping Τ is established by solving a set of scalar equations. Periodic difference equations of extended Ricker type and also rational difference equations with a finite number of delays are also considered by reducing them to equations without delays but with a larger period. Conditions are given guaranteeing the existence and global asymptotic stability of periodic solutions.     

Convergence in almost periodic Fisher and Kolmogorov models

 We study convergence of positive solutions for almost periodic reaction diffusion equations of Fisher or Kolmogorov type. It is proved that under suitable conditions every positive solution is asymptotically almost periodic. Moreover, all positive almost periodic solutions are harmonic and uniformly stable, and if one of them is spatially homogeneous, then so are others. The existence of an almost periodic global attractor is also discussed. Received: 11 November 1996 / Revised version: 8 January 1998     

Periodic solutions to nonautonomous difference equations

A technique is presented for determining when periodic solutions to nonautonomous periodic difference equations exist. Under certain constraints, stable periodic solutions can be guaranteed to exist, and this is used to compare the analogous behavior of a nonautonomous periodic hyperbolic difference equation to that of the nonautonomous periodic Pearl-Verhulst logistic differential equation.     

Two species competition in a periodic environment

The classical Lotka-Volterra equations for two competing species have constant coefficients. In this paper these equations are studied under the assumption that the coefficients are periodic functions of a common period. As a generalization of the existence theory for equilibria in the constant coefficient case, it is shown that there exists a branch of positive periodic solutions which connects (i.e. bifurcates from) the two nontrivial periodic solutions lying on the coordinate axes. This branch exists for a finite interval or spectrum of bifurcation parameter values (the bifurcation parameter being the average of the net inherent growth rate of one species). The stability of these periodic solutions is studied and is related to the theory of competitive exclusion. A specific example of independent ecological interest is examined by means of which it is shown under what circumstances two species, which could not coexist in a constant environment, can coexist in a limit cycle fashion when subjected to suitable periodic harvesting or removal rates.Research supported by National Science Foundation Grant No. MCS-7901307