时滞微分方程的周期解问题

项目名称： 时滞微分方程的周期解问题

项目编号： No.11471085

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 庾建设

作者单位： 广州大学

项目金额： 65万元

中文摘要： 周期解理论是运动周期轨道的存在性和稳定性的理论。对于不能求解的运动方程寻找其周期解并研究其稳定性一直是微分动力系统的一个重要研究课题。本项目研究时滞微分方程周期解的存在性、唯一性、多重性和稳定性问题。主要内容包括：一、时滞周期运动的最小周期问题，即如何证明周期轨道周期的最小性以及周期解的唯一性和具同一周期的轨道的多重性问题；二、时滞周期运动周期解的稳定性问题；三、时滞微分方程在分子生物学数学模型的应用。关于时滞微分方程周期解的研究成果虽然很多，但涉及给定最小周期的周期解存在性、唯一性和多重性的研究却相当少，远未形成系统的理论。而具有特殊对称群的Hamilton系统与时滞微分方程有密切的关系。因此本项目计划综合应用时滞微分方程理论，Fourier分析方法，临界点理论以及现代非线性泛函分析等工具对以上内容进行系统研究，获得创新性强的研究成果，然后将研究成果应用于具体的分子生物学数学模型中。

中文关键词： 周期解；时滞微分方程；临界点理论；最小周期；哈密顿系统

英文摘要： Periodic solution theory mainly concerns with the existence and stability of periodic orbits. A very important and significant topic is to find periodic solutions to differential equations which can not be solved explicitly and to study their stability. This project mainly investigate the existence, uniqueness, multiplicity and stability of periodic solutions to delay differential equations. It consists of three parts. First, existence of periodic solutions of delay differential equations with prescribed minimal period, including the minimality of the period of periodic solutions, uniqueness of periodic solutions and multiplicity of periodic orbits with the same period; Second, stability of periodic sultions to delay differential equations； And finally, applications of delay differential equations to mathematical models derived from molecular biology. There have been a lot of works on periodic solutions to delay differential equations, only a few results are concerned with minimality of period, uniqueness and multiplicity of periodic solutions. No systematic theory and methods are built to deal with such problems. Nevertheless, there is a close relationship between Hamilton systems with special symmetric groups and delay differential equations. Therefore, the purpose of this project is to apply comprehensively the theory of delay differential equations, Fourier Analysis, critical point theory and modern nonlinear functional analysis to study systematically these topics,so as to obtain creative results, and then to apply them to the study on the mathematical models in molecular biology.

英文关键词： periodic solution;delay differential equation;critical point theory;minimal period;Hamitonian system