非线性微分方程的一些问题

This project is mainly concerned with some contents in dynamical systems，such as almost periodic motions, bifurcations, ionic flow, attractor, stability, travelling waves, homoclinic orbit, and so on. We would like to mainly focus on several significant types of differential euqations or interesting mathematical models, and study their dynamical properties for these equations or models. The equations we will study mainly are age-structured equations, delay differential equations, reaction diffusion equations with delay. The models we will study mainly are biological mathematical models. The approach we will use mainly is to use the theory of dynamical systems, the qualitative theory of differential equations, and nonlinear functional analysis. We will improve and provide the models according to some practical problems. We will bigin and do our reasearch works according to the characteristic of the models. According to the types of equations and mathematical models, we will study these problems needed to be solved in the theory of dynamical systems and differential equations. This project will intend to enrich the theory of dynamical systems and differential equations corresponding to our problems, and provide some interesting mathematical explainations for some models.

本项目主要涉及动力系统中的如下几个方面的内容：概周期运动，分支问题，离子流，吸引子，稳定性，行波解，同宿轨，等问题。 我们主要想考虑几类重要的微分方程类型或重要的数学模型，研究这些方程类型或模型的动力学性质。研究的方程类型主要是年龄结构方程，时滞微分方程，时滞反映扩散方程， 数学模型主要是生物数学模型。主要方法是运用动力系统理论，微分方程定性理论，和非线性分析理论。 针对实际问题提出模型，根据模型特征从事研究工作。 通过方程类型和数学模型提出并研究动力系统理论和微分方程理论需要解决的问题。 我们的研究将丰富相关的动力系统理论和微分方程理论，并为实际问题提供数学理论依据。

该项目研究了概周期微分方程，分支与稳定性，局部与非局部方程的行波解，有变分结构方程的同宿解, 离子流，等方面的内容。讨论了生态学中的动力学模型和传染病中的动力学模型，考虑了局部扩散和非局部扩散，研究了行波解的存在性和稳定性，正平衡点的存在性、稳定性和分支。利用非稠定发展方程的中心流形理论，研究了年龄结构方程的Hopf分支和B-T分支。研究了重合度理论与概周期微分方程概周期解的存在性问题，提供了用重合度理论研究概周期解的可行性方法。用变分法和临界点理论，讨论了二阶Hamilton系统，分数阶的 Hamilton系统，非自治的四阶系统，变指数的Hamilton系统，研究了同宿解的存在性。 讨论了离子流Poisson-Nernst-Planck方程解的存在性。