脉冲时滞微分方程的周期解及数值计算问题研究

项目名称： 脉冲时滞微分方程的周期解及数值计算问题研究

项目编号： No.11501193

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 张丹

作者单位： 东莞理工学院

项目金额： 18万元

中文摘要： 脉冲时滞微分方程在很多领域具有广泛的应用，如连续力学、种群生态学、电子学、核反应堆动力学及现代控制论等等，研究脉冲时滞微分方程的理论和应用具有非常重要的意义。到目前为止，许多专家学者对脉冲时滞微分方程的周期解、边值问题等方面进行了广泛地研究，但采用的方法通常是不动点理论，而用临界点理论研究脉冲时滞微分方程的文献很少。其次，也很少有学者在研究周期解存在性的同时，考虑此类方程的数值解及其稳定性和收敛性问题。 本项目旨在利用临界点理论研究几类脉冲时滞微分方程周期解的存在性与多解性问题，突出由脉冲生成周期解的问题研究，揭示时滞和脉冲扰动对周期解的实质影响，并利用几何指标理论对解的个数进行估计，同时对脉冲时滞微分方程周期解的数值计算及其稳定性与收敛性问题进行研究。

中文关键词： 脉冲时滞微分方程；周期解；数值解；临界点理论；变分方法

英文摘要： Impulsive delay differential equations are applied in many fields, such as continuous mechanics, population ecology, electronics, nuclear reactor dynamics and modern control theory and so on. It is of very important significance to study the theory and applications of impulsive delay differential equations. So far, the periodic solutions and the periodic boundary value problems of Impulsive Differential Equations with delay had been extensively studied by many experts and scholars, but adopted method is usually fixed point theory. However, the reference about studying impulsive delay differential equations via critical point theory is rarely. In addition, few scholars considered the numerical solutions and the convergence and stability of solutions for impulsive delay differential equations while studying the existence of periodic solutions for these equations. The purpose of this project is to research the existence and multiple of periodic solutions for several classes of impulsive delay differential equations using critical point theory, to outstand the research of periodic solutions generated by impulses, and to reveal the actual impact of the time delay and impulse disturbance on periodic solutions. The number of solutions will be estimated by the geometrical index theory, and the numerical calculation of periodic solutions for impulsive delay differential equations and their stability and convergence will also be studied.

英文关键词： impulsive delay differential equations ;periodic solution; numerical solution;critical point theory;variational method