几类时滞微分方程和扰动微分系统解的定性研究

Delay differential equations and singularly perturbed differential systems are very interesting subjects, which have significant practical background. Based on the variational method, nonlinear analysis and geometric singular perturbation theory, we study the following problems:. Firstly, we study some nonlinear differential equations with multiple delays and obtain the existence of solutions, multiple solutions and the existence of infinite solutions, the relationship between the number of solutions and the size of the delay, and the bifurcation of the delay as a parameter. Secondly, we will discuss the stability, the existence of local and global solutions and the existence, stability and nonexistence of periodic solutions for a class of predator-prey systems with multiple delays. Lastly, we study some singularly perturbed reaction-diffusion equations and differential systems. We will obtain the existence of homoclinic orbits and heteroclinic orbits connecting equilibrium and travelling wave solutions, asymptotic behavior of these differential systems.. This project will play an important role in the development of the study of delay differential equations, singular perturbation theory and applications. It is significant.

时滞微分方程和扰动微分系统是一个非常有意义的研究课题，有着重要的实际背景。本项目主要运用变分法、非线性分析理论和几何奇异摄动理论等研究以下问题：. 1、研究含有多时滞微分方程的解的存在性、多解性和无穷多解的存在条件，解的个数与时滞量大小的关系以及时滞量作为参数时的分叉现象；2、研究带有多时滞的多种群捕食竞争系统的稳定性、局部和全局解的存在性以及周期解的存在性、稳定性和不存在性等；3、研究带有扰动的反应扩散方程和微分系统平衡点之间的同宿轨道和异宿轨道的存在性、行波解的存在性和渐近性。. 本项目将对时滞微分方程、奇异摄动理论及其应用的研究发展起促进作用，具有重要的意义。

本项目主要研究带有时滞的波方程和种群动力系统的动力学性质，非线性波动方程及其行波解的研究在流体力学等领域具有非常重要的意义，种群动力系统和反应扩散方程的动力学研究, 对生态系统中的很多变化趋势和规律可以进行合理的解释,有着重要的意义。本项目研究带有扰动的反应扩散方程、时滞Camassa-Holm方程和Camassa-Holm-KP方程；带有非局部时滞的两种群捕食与被捕食反应扩散系统、具有功能反应函数的Gause型捕食模型；带有时滞的色散-耗散方程；三阶非线性奇异摄动微分系统和边值问题，Cohen-Grossberg型双向联想记忆的神经网络等；得到了行波解和孤波解的存在性、同异宿轨道的稳定性、行波波速的单调性和极限等动力学性质。项目组在Journal of Functional Analysis, Journal of Differential Equations，Qualitative Theory of Dynamical Systems等期刊上发表论文18篇，其中1篇论文被Web of Science列为ESI高被引论文，共培养16名研究生。本项目对非线性时滞微分方程和奇异摄动理论及其应用的研究发展起促进作用，具有重要的意义。