Solution structure around bifurcation points of co-dimension 2

Masaharu Nagayama（本研究项目的研究人员之一）设计了一种计算机代码，可以分析双分岔点附近的分岔结构。该程序处理单稳定反应扩散系统脉冲解的分岔现象，并具有以下两个功能：(1)找到临界点并构造其分岔分支；(2)扩展已有的分岔分支。为了设计代码，我们考虑了有限区间（0，L）上的反应扩散系统，并满足周期边界条件，其中L是一个固定的大正数。从相位条件出发，我们得到了决定脉冲传播速度的方程，并用Keller方法表示了方程系统中对参数p的依赖关系。在计算机代码中，用牛顿法对上述公式化的问题进行了数值求解。我们注意到解是{反应扩散系统的解，c， p}的集合。当一个行脉冲从一个驻脉冲分叉时，会出现两个零特征值，其中一个是平凡的，一个是平凡的，对应于平行。我们的代码不仅适用于这种情况，而且适用于除了平凡的情况外存在两个关键零特征值的情况。首席研究员处理了自传播高温合成数学模型的静止和移动燃烧脉冲，包括冷却效果和原料供应系统。我们采用分段常数函数作为反应项，从数学上严格地研究了脉冲解的存在性，以及燃烧脉冲在圆域上的碰撞动力学。

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.