Studies on Asymptotic Properties of Solutions of Differential Equations

这项研究的目的是检验1.Lienard系统轨迹的渐近行为和2.微分-差分方程组的稳定区域。确定(1)Lienard系统的轨迹是否接近原点和(2)轨迹是否与垂直等斜线相交的条件，对于解的振动和其他学科是非常重要的。本研究的首席研究员和合作者对上述问题进行了讨论，得到了一些隐式的充要条件、显式的必要条件和显式的充分条件(非线性差等式应用)。作为结果，我们澄清了Lienard平面上分离阵的性质，给出了具有同宿轨的Lienard系统周期解的存在性结果，这是分离线(Disc.Cont.Dynam.Syst)的一个典型例子。我们还给出了Lienard系统的全局相图按性质和分离矩阵的数目的完全分类(J.Math.Anal.Appl)。此外，我们还讨论了一类二阶欧拉型非线性微分方程(Proc.amer.Math.Soc.)和周期性强制的Lienard系统(Proc.Inter.Con.Dynam.Syst.Chaos，Diff.Integ.Eq.)。作为对生物模型的应用，我们考虑了一类具有一般Holling功能反应的捕食者-食饵系统，给出了该系统恰好存在一个稳定极限环的充要条件。给出了一类n维微分-差分系统零解指数渐近稳定的充要条件。将拟多项式的Pontryagin判据应用于微分-差分方程组的特征方程，证明了我们的结果。当n=1时，我们的结果给出了标量微分差分方程解的一个众所周知的充要条件。

The purpose of this research is to examine 1. asymptotic behavior of trajectories of the Lienard system and 2. stability regions for systems of differential-difference equations.1. It is very important for oscillation of solutions and other subjects to find conditions for deciding (1) whether trajectories of the Lienard system approach the origin or not and (2) whether trajectories intersect the vertical isocline. The head investigator and cooperators of this research discussed the above problems and obtained some implicit necessary and sufficient conditions, explicit necessary conditions and explicit sufficient conditions (Nonlin.Diff.Eq.Appl.). As the result, we clarified the properties of separatrices in the Lienard plane and gave existence results on periodic solutions of the lienard system with a homoclinic orbit which is a typical example of separatrix (Disc.Cont.Dynam.Syst.). We also gave a complete classification of global phase portraits of the Lienard system by the properties and the number of separatrices (J.Math.Anal.Appl.). Moreover, we discussed oscillation problems of a second order nonlinear differential equation of Euler type (Proc.Amer.Math.Soc.) and a periodically forced Lienard system (Proc.Inter.Con.Dynam.Syst.Chaos, Diff.Integ.Eq.). As an application to biological model, we considered a predator-prey system with a fairly general functional response of Holling type and presented a necessary and sufficient condition under which this system has exactly one stable limit cycle.2. We gave some necessary and sufficient conditions for the zero solution of an n-dimensional differential-difference system to be exponential asymptotically stable (Funk.Ekvac.). The proof of our results is carried out by an application of Pontryagin criterion for quasi-polynomials to the characteristic equation of the differential-difference system. In case n=1, our results yeild a well-known necessary and sufficient condition for a scalar differential-difference equation.