时滞微分方程若干余维2分支问题的数值方法研究

项目名称： 时滞微分方程若干余维2分支问题的数值方法研究

项目编号： No.11201061

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 徐英祥

作者单位： 东北师范大学

项目金额： 22万元

中文摘要： 微分方程的余维2分支是探索高余维分支的基础，其数值方法研究对分支计算软件的开发、复杂非线性系统的数值模拟等具有重要意义。然而，时滞微分方程余维2分支的数值计算要面临来自方程的无穷维相空间以及分支的高度退化性的双重困难。本项目将具体刻画时滞微分方程在余维2奇点处相应于临界特征值的广义特征空间，并据此将无穷维空间中的问题等价转化至有限维空间考虑。利用中心流形约化与规范型理论等工具，结合正则化技巧，针对时滞微分方程余维2分支问题的数值计算具体开展如下工作：a)余维2奇点计算的算法设计与理论分析；b)余维2奇点处解支转接算法设计及理论分析；c)数值离散对连续方程余维2分支局部分支结构的保持性分析与判定。项目研究将进一步丰富和拓展时滞微分方程分支计算的相关算法与理论，实践上可为自动控制、生物数学、新材料研发等相关领域分支问题的数值仿真提供方法支持与理论保障。

中文关键词： 数值方法；分支问题；余维2；时滞微分方程；保持性

英文摘要： The codimension 2 bifurcations of differential equations are the foundations of investigating the bifurcations of higher codimension. Their numerical analysis is of great importance for developing the bifurcation softwares and simulating the complex nonlinear systems. However, the numerical analysis for codimension 2 bifurcations of delay differential equations encounters grave difficulties from both the infinite dimensionality of the phase space and the high degeneracy of the bifurcations. The project shall describe specifically the generalized eigenspace associated with the critical eigenvalues at the singular point and, based on the descriptions above, turn the problems of infinite dimension into the equivalent ones of finite dimension. By use of the tools like center manifold reduction and normal form theory, jointing with the regularization techniques, we will carry out the following works concerning the numerical analysis for codimension 2 bifurcation problems of delay differential equations: a) algorithms and theories for computing the singular points of codimension 2; b) algorithms and theories for branch switching near the singular points of codimension 2; c) preservation of bifurcation structures for codimension 2 bifurcations by numerical discretization. The research will enrich and broaden the theori

英文关键词： Numerical methods；Bifurcation problems；Codimension 2；Delay differential equations；Preservation