Understanding of Spatio-temporal patterns by Singular Limit Methods

奇异极限法的本质是从复杂动力学中提取骨架结构。一个典型的例子是一类利用时空尺度的差异从偏微分方程中导出的界面方程。所得方程继承了原动力学的基本部分。我们的项目不仅将其有效性扩展到各种系统，而且为复杂动力学提供了一种新的方法，例如自我复制动力学和自毁过程，这些被认为是瞬态动力学，因此仍然是开放的问题。我们方法的亮点有三个方面：(a)驱动复杂动力学的全局分岔分支集的几何表征。(b)脉冲-脉冲相互作用方法，阐明了复杂动力学的开始以及局部模式之间的弱相互作用。(c)组织高共维中心，这解释了微型复杂动力学的起源。要解决上述问题，借助于AUTO等路径跟踪软件的数值方法是必不可少的，尤其是(a)。这些方法以一种协调的方式很好地理解了复杂动力学的各个方面，并具有应用于其他领域的巨大潜力。主要研究者西浦康正于2002年被日本数学会授予“秋季奖”。

The essence of the singular limit method is to extract a skeleton structure from complex dynamics. A typical example is a class of interfacial equations derived from PDEs by using differences of space-time scales. The resulting equations inherit the essential parts of original dynamics. Our project not only extend its validity to a variety of systems but also present a new approach to complex dynamics, for instance self-replicating dynamics and self-destruction process, which have been regarded as transient dynamics, therefore remain as open questions. The highlight of our approach is three-fold : (a) Geometric characterization for the set of global bifurcating branches that drives complex dynamics. (b) Pulse-pulse interaction method which clarifies the onset of complex dynamics as well as weak interaction among localized patterns. (c) Organizing centers of high codimensions which explains an origin of complex dynamics in a miniature size.Numerical approach with the aid of path-tracking software like AUTO is indispensable to accomplish the above issues, especially (a). These methods works very well in a harmonized way to understand the whole aspects of complex dynamics and has a great potential to be applied to other fields. The main investigator Yasumasa Nishiura was awarded "Autumn Prize" by Japan Mathematical Society in 2002.