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Bifurcation of traveling spot patterns in 2-dimensional domain

二维域中行进光斑图案的分叉

基本信息

批准号：
09640252
负责人：
IKEDA Hideo
金额：
$ 1.73万
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：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

Bifurcation problem of traveling spot patterns in 2-dimensional domain is studied. Our model equations are 2- component reaction-diffusion systems including a non-local term. . Numerically they exhibit traveling spot patterns which is bifurcated from standing spot patterns stably. Theoretically we know that 2-component reaction-diffusion systems without a non-local term do not exhibit such phenomena. Then first we considered that such phenomena came from the non-local term. But later we could succeed in rewriting the 2-component systems with the non-local term into the 3-component reaction-diffusion systems without a non-local term. We conclude that such phenomena come from the delicate balance of the ratio of diffusions, the ratio of reactions and the nonlinear terms. This enables us to analyze them mathematically. In 1-dimensional case, we showed the existence and the stability of traveling front and back solutions under the assumption of bistability. Now we try to construct standing pulse solutions connecting these traveling front and back solutions. For the stability property, we will show that standing pulse solutions destabilize under the out-of-phase (asymmetric) modes first and the in-phase (symmetric) modes secondly when some parameter is changed. At this time, the informatin of the stability of traveling front and back solutions and the connecting manner of these two solutions will help us to analyze the stability property. These new bifurcated solutions are stable traveling pulse solutions. In 2-dimensional case, it is confirmed numerically that these systems have stable 2-dim traveling spot patterns. For this case, we will show the existence of radial symmetric standing pulse solutions and then catch traveling spot patterns as the destabilization of radial symmetric standing pulse solutions.

研究了二维区域中行斑图的分岔问题。我们的模型方程是包含非局域项的2组分反应扩散系统。在数值上，它们表现出稳定的从静止斑模式分岔的移动斑模式。理论上我们知道，没有非局域项的双组分反应扩散系统不会出现这种现象。然后，我们首先考虑了这种现象来自于非本地术语。但后来我们可以成功地把有非局部项的两组分系统改写成没有非局部项的三组分反应扩散系统。我们得出结论，这种现象来自扩散比、反应比和非线性项的微妙平衡。这使我们能够用数学方法分析它们。在一维情况下，我们证明了在双稳定假设下前后移动解的存在性和稳定性。现在我们试着构造固定脉冲解连接这些前后移动的解。对于稳定性，我们将证明当某些参数改变时，驻脉冲解首先在非对称模式下失稳，其次在对称模式下失稳。此时，获取前后解的稳定性信息以及前后解的连接方式将有助于分析其稳定性。这些新的分岔解是稳定的行脉冲解。在二维情况下，数值证实了这些系统具有稳定的二暗淡行斑模式。对于这种情况，我们将证明径向对称驻脉冲解的存在性，然后捕捉作为径向对称驻脉冲解的不稳定的行进斑点图案。