Solution structure around bifurcation points of co-dimension 2

余维 2 分叉点周围的解结构

基本信息

批准号：
15340038
负责人：
IKEDA Tsutomu
金额：
$ 4.8万
依托单位：
Ryukoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

Masaharu Nagayama (one of investigators of the present research project) has devised a computer code that can analyze bifurcation structures in a neighborhood of double bifurcation points. This code deals with bifurcation phenomena of pulse solutions to mono-stable reaction-diffusion systems, and is equipped with the following two functions : (1)It can find a critical point and construct its bifurcation branch, (2)It can extend existing bifurcation branches. In order to devise the code, we consider the reaction-diffusion system on a finite interval (0,L) subject to the periodic boundary condition where L is a fixed large positive number. From the phase condition we obtain the equation that determines the propagating velocity of traveling pulse, and by the Keller method we express the dependency on a parameter p included in the equation systems. The problem formularized as in the above is numerically solved by the Newton method in the computer code. We note that a solution is a set of {solutions to reaction-diffusion systems, c, p}. When a traveling pulse bifurcates from a standing pulse, there appear two zero-eigenvalues, one of which is a trivial one trivial one corresponding to parallel translation. Our code applies to not only this case but also the cases where two crucial zero-eigenvalues exist except the trivial one.The head investigator have dealt with standing and traveling combustion pulses of a mathematical model for self-propagating high-temperature syntheses including both the cooling effect and raw material supply system. Employing a piece-wise constant function for the reaction term, we have studied the existence of pulse solutions in a mathematically rigorous way, and also the collision dynamics of combustion pulses on a circle domain.

Masaharu Nagayama（本研究项目的研究人员之一）设计了一种计算机代码，可以分析双分叉点附近的分叉结构。该代码处理单稳态反应扩散系统脉冲解的分岔现象，具有以下两个功能：（1）找到临界点并构造其分岔分支，（2）扩展现有分岔分支机构。为了设计代码，我们考虑有限区间 (0,L) 上的反应扩散系统，该系统服从周期性边界条件，其中 L 是固定的大正数。根据相位条件，我们得到了确定行进脉冲传播速度的方程，并通过凯勒方法，我们表达了对方程组中包含的参数 p 的依赖性。上述公式化的问题通过计算机代码中的牛顿法进行数值求解。我们注意到，解是一组{反应扩散系统的解，c，p}。当行进脉冲与驻波脉冲分叉时，会出现两个零特征值，其中一个是对应于平行平移的平凡一平凡一。我们的代码不仅适用于这种情况，还适用于存在两个关键零特征值（除了平凡的零特征值）的情况。首席研究员处理了用于自传播高温合成的数学模型的驻留和行进燃烧脉冲，包括冷却效果和原料供应系统。采用分段常数函数作为反应项，我们以数学上严格的方式研究了脉冲解的存在性，以及圆域上燃烧脉冲的碰撞动力学。