Mathematical Theory of Nonlinear-Non-equilibrium Reaction-Diffusion Systems

非线性非平衡反应扩散系统的数学理论

基本信息

批准号：
18104002
负责人：
MIMURA Masayasu
金额：
$ 45.09万
依托单位：
Meiji University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (S)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2010
项目状态：
已结题

项目摘要

Reaction diffusion equations already appeared as mathematics models which described population genetics, ecology and so on from the early 20^<th> century, and the qualitative study was performed in the world of mathematics. In the late 20^<th> century, there was a great breakthrough in sciences, that is, nonlinear non-equilibrium science and reaction diffusion equations appeared in natural ecienc such as physics, chemistry, biology and other fields, as mathematics models describing various nonlinear non-equilibrium phenomena. Thus, the study of reaction diffusion system has been pushed forward not only in mathematics but also in widely natural sciences. The result in this study was able to establish the analytical technique of spatio-temporal patterns arising in reaction diffusion equations from viewpoints of mathematics and applied mathematics for mathematical elucidation of the nonlinear-non-equilibrium phenomenon. As examples, there are (1) the construction of "the invariant manifold theory of infinite dimension dynamical systems" to handle pattern dynamics appearing as dissipative structure and self-organization, (2) the construction of "analytical theory of transient pattern dynamics" to understand the transition process from a simple pattern to a complex one, which is a typical nonlinear-non-equilibrium phenomenon, and (3) the construction of "the singular limit theory" to understand complex dynamic patterns and stationary forms in nonlinear non-equilibrium systems. These results enable us to mathematically understand spatio-temporal patterns in nonlinear non-equilibrium systems.

反应扩散方程已经显示为数学模型，描述了20个世纪初期的人口遗传学，生态学等，而定性研究是在数学领域进行的。在20个世纪后期，科学的突破很大，也就是说，非线性的非平衡科学和反应扩散方程出现在自然环保中，例如物理学，化学，生物学和其他领域，如数学模型，描述了描述各种非平衡现象的数学模型。因此，对反应扩散系统的研究不仅在数学中，而且在广泛的自然科学中都被推进了。这项研究的结果能够建立从数学观点和应用数学的反应扩散方程中产生的时空模式的分析技术，用于数学阐明非线性 - 非平衡现象。作为示例，有（1）构建“无限尺寸动态系统的不变流形理论”，以处理出现的模式动力学，以耗散性结构和自组织形式出现，（2）构建“瞬态模式动态的分析理论”，以理解从简单的模式到复杂的模式，这是一个典型的非线性 - 非线性的范围，这是一个典型的范围，这是一个典型的范围，这是一个典型的界限理论。非线性非平衡系统中的动态模式和固定形式。这些结果使我们能够在非线性非平衡系统中数学上了解时空模式。