时滞微分方程若干余维2分支问题的数值方法研究

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 theories of numerical methods for bifurcation problems of delay differential equations, and further, provide the methodological and theoretical support for the numerical simulation of bifurcation problems applied in control systems, biomathematics and new materials development, etc.

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。

微分方程的余维2分支是探索高余维分支的基础，其相关数值方法研究对分支计算软件的开发、大规模复杂非线性系统的数值模拟等具有重要意义。本项目以具有余维2奇点时滞微分方程的相关数值算法研究以及数值离散化对余维2分支结构的保持性分析为主要研究内容，项目研究取得如下主要成果:1.提出了新的Takens-Bogdanov点和Double Hopf点的数值计算方法，并分析了数值方法的有效性；2.证明了欧拉方法、theta方法等数值方法保持时滞微分方程Takens-Bogdanov分支以及fold-Hopf分支局部分支结构，为数值模拟时滞微分方程的分支结构提供了理论保证；3.分析了具时滞反馈可饱和吸收激光器模型的Takens-Bogdanov分支，给出了分支产生的条件并获得了参数平面上局部分支结构表达；4.分析了大规模微分线性系统的快速计算方法，为复杂非线性系统的数值模拟奠定了理论基础。