时滞微分与差分系统的周期性以及同宿异宿轨问题

项目名称： 时滞微分与差分系统的周期性以及同宿异宿轨问题

项目编号： No.10871053

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 郭志明

作者单位： 广州大学

项目金额： 27万元

中文摘要： 本项目主要研究变分方法在时滞微分与差分系统周期解、同宿轨和异宿轨问题中的应用。具体说来，对含有时滞的微分与差分系统寻求适当的函数空间和含有偏差变元的作用泛函，应用和发展临界点理论中Minimax方法、几何指标理论、Morse理论与Maslov指标理论等工具，建立Hilbert空间上含有偏差变元泛函的临界点存在性定理与多解性定理，并用来研究时滞微分与差分系统周期解与次调和解的存在性、多解性以及最小周期等问题；进一步研究时滞微分系统与差分系统的同宿轨与异宿轨的存在性；将Kaplan-Yorke型方程的有关结果推广到非自治以及高维情形；开展对时滞微分与差分系统的应用研究，对人口动力学、经济学及自动控制中出现的各类时滞模型的周期振荡进行系统的研究，揭示其内在的本质规律。这些成果将为研究时滞微分与差分系统提供一种新的途径，填补这一领域的研究空白。这项研究既具有重要的理论意义又具有广阔的应用价值。

中文关键词： 时滞微分方程；差分方程；周期解；同宿异宿轨；变分方法

英文摘要： This project mainly concerns with the applications of variational methods to the periodic solutions, homoclinic orbits and heteroclinic orbits of delay differential equations and delay difference equations. More specifically, suitable function spaces and action functionals defined on them with deviating arguments will be introduced for differential equations and difference equations with delays. By developing and applying the minimax methods, geometrical index theories, Morse theory and Maslov theory in the critical point theory, some existence theorems and multiplicity results will be established for critical points of functionals with deviating arguments defined on Hilbert spaces, which will be used to study the existence, multiplicity and minimal period problems for periodic solutions and subharmonic solutions to delay differential equations and difference equations. Furthermore, the existence of homoclinic and heteroclinic orbits will also be investigated for delay differential equations and difference equations. The results on Kaplan-Yorke type equation will be generalized to the cases of nonautonomous equations and high-dimensional equations. Applications of the theories of delay differential equations and difference equations will be studied systematically to the periodic oscillations of mathematical models with delays, which appear in the study of population dynamics, economics and automation, etc. These researches will provide a new approach to study delay differential systems and difference systems and will fill a gap in this field. To sum up, this project is of important significance both theoretically and practically.

英文关键词： delay differential equation; difference equation; periodic solution; homoclinic and heteroclinic orbits; variational method