Solution structure around bifurcation points of co-dimension 2

长山正治(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。该程序研究单稳态反应扩散系统脉冲解的分支现象，具有如下两个功能：(1)寻找临界点并构造其分支；(2)扩展已有的分支。为了设计程序，我们考虑了有限区间(0，L)上满足周期边界条件的反应扩散方程组，其中L是固定的大正数。从位相条件出发，我们得到了确定行波脉冲传播速度的方程，并用Keller方法表示了它对方程组中参数p的依赖关系。通过计算机代码中的牛顿法对上述公式化问题进行数值求解。我们注意到解是反应扩散系统的{解，c，p}的集合。当行波脉冲与驻波脉冲分叉时，会出现两个零本征值，其中一个是平移对应的平凡本征值。我们的代码不仅适用于这种情况，而且还适用于除了平凡的零本征值之外存在两个关键零本征值的情况。首席研究员处理了包括冷却效应和原料供应系统在内的自传播高温合成数学模型的驻留和行进燃烧脉冲。采用分段常数函数作为反应项，用严格的数学方法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。

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.