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On the existence and stability of singularly perturbed solutions for one-activator and two-inhibitors reaction-diffusion models

关于一激活剂和二抑制剂反应扩散模型奇异摄动解的存在性和稳定性

基本信息

批准号：
13640107
负责人：
IKEDA Hideo
金额：
$ 0.96万
依托单位：
Toyama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

项目摘要

Our purpose of this project is to find a stationary solution of a three-component reaction-diffusion system and examine the stability of the solution by applying a singular perturbation method. The most difficult problem in these process is to solve a reduced problem, that is, to find a C^1'-class solution of the pair of elliptic boundary value problems with discontinuous nonlinearities and to know the dependency of the solution on physical parameters. With a dynamical system view point, the reduced problem is rewritten as a four-dimensional dynamical system with discontinuous nonlinearities. There are two hyperbolic equilibrium points, both have two-dimensional stable manifolds and two-dimensional unstable manifolds, in R^4. The above problem corresponds to find a continuous trajectroy on the restricted two-dimensional space R^2. Under an artificial condition, we succeed in finding such trajectroy. And using this, we show the existence of a solution of the original problem applying a singular perturbation method. Next we consider the same problem on the two-dimensional region R^2. Under a similar condition to the above, we can find a radialy symmetric solution. Finally we study the linearized eigenvalue problem around the solution and trace the, critical eigenvalue by numerical simulation. We check the appearance of instability modes, that is, the 0-th mode and the first mode. The former case corresponds to the bifurcation phenomena of a travelling wave solution, that is, a traveling spot, from a standing wave solution and the latter case does to the bifurcation of an asymmetric solution from a radialy symmetric standing wave solution. But we cannot find a parameter on which the Hopf bifurcation occurs.

本计画的目的是利用奇异摄动法来寻找一个三组份反应扩散系统的定态解，并检验解的稳定性。其中最困难的问题是求解一个约化问题，即求出一对具有间断非线性项的椭圆边值问题的C^1 '类解，并知道解对物理参数的依赖性。从动力系统的观点出发，将简化问题转化为具有间断非线性项的四维动力系统。在R^4中有两个双曲平衡点，它们都有二维稳定流形和二维不稳定流形。上面的问题对应于在受限的二维空间R^2上找到一个连续矩阵。在人为的条件下，我们成功地找到了这样的缺陷。并利用这一点，我们显示的存在性的解决方案的原始问题应用奇异摄动方法。接下来我们考虑二维区域R^2上的同样问题。在类似的条件下，我们可以找到一个径向对称的解决方案。最后，我们研究了解附近的线性化特征值问题，并通过数值模拟追踪了临界特征值。我们检查的不稳定模式，即第0模式和第一模式的出现。前一种情况对应于行波解从驻波解分支出来的现象，即行波点从驻波解分支出来的现象，后一种情况对应于非对称解从径向对称驻波解分支出来的现象。但我们找不到一个参数上的霍普夫分岔发生。