多时滞微分系统的余维分支分析及应用

By using the bifurcation theory of delay differential equations to study the higher codimension bifurcations such as Bogdanov-Takens (B-T) bifurcation and triple-zero bifurcation and so on for some differential systems with multiple delays is an important part of bifurcation theory of differential equations. The core idea of the study is dimensionality reduction, the main methods used to research the bifurcation properties of some systems are the center manifold theorem and normal form theory: according to different codimension number of the systems at their singularities, one can deduce different normal forms, by analyzing the properties of the normal forms, the corresponding properties of the systems near singularity can be acknowledged. The main contents of this project are as follows: 1. by using the coefficient formulae deduced by us of second-order normal form of B-T or triple zero bifurcation about some differential systems with more delays, the properties of some high dimension neural network models or oscillator systems will be further studied; 2. we will deduce the third order normal forms coefficient formulae of codimension two or three bifurcations for some differential systems with multiple delays; 3. we will analyze the codimension 2 or 3 bifurcation properties of some biomathematics models with multiple delays. These problems have been paid extensive attention by experts from domestic and overseas, our deep study will develop and enrich the existing bifurcation theory of differential equations.

应用时滞微分方程分支理论分析多时滞微分系统的高余维分支如Bogdanov-Takens分支(简称B-T分支)和triple-zero分支等是微分方程分支理论的重要组成部分并具有十分重要的实际意义。研究的核心思想是降维，在分析系统的分支性质时主要用到中心流型定理和标准型理论：根据系统在奇点处不同的余维数得到不同的规范型，由规范型与原系统在奇点附近性质的等价性知原系统的性质。本课题主要研究内容：1. 应用我们已经推导出的多个时滞微分系统的B-T分支和triple-zero分支的二次规范型的系数公式来研究一些高维的神经网络模型或振动系统的性质；2. 推导出具有多时滞微分系统的余维2或3分支的三次截断规范型的系数公式；3. 分析一些具有多个时滞的生物数学模型的余维2或3分支性质。这些问题是国内外学者共同关注的热点，我们的深入研究能够发展和丰富已有的微分方程分支理论。

本项目主要结果有三个方面：(1) 应用微分方程的稳定性理论、中心流形定理和标准型理论分析几类多时滞神经网络模型和捕食被捕食模型(也称为泛函微分方程)在某个平衡点处的B-T分支、triple zero分支等性质；(2) 理论推导出了Turig-Hopf分支参数范围来分析偏微的捕食被捕食模型的性质；(3) 侧重于数值模拟分析多个正平衡点的奇点类型的具有进化理论和阿利效应的种群模型的性质。所得公式及分析思路可以用来分析更多的相关模型的动力学性质。