Understanding of Spatio-temporal patterns by Singular Limit Methods

奇异极限法的实质是从复杂动力学中提取骨架结构。一个典型的例子是利用时空尺度差从偏微分方程导出的一类界面方程。由此产生的方程继承了原始动力学的基本部分。我们的项目不仅将其有效性扩展到各种系统，而且还为复杂动力学提供了一种新的方法，例如自复制动力学和自毁过程，这些动力学一直被认为是瞬态动力学，因此仍然是开放的问题。我们的方法的亮点是三重：（a）几何表征的一组全球分叉分支，驱动复杂的动态。(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域模式之间的弱相互作用。(c)组织高余维的中心，这解释了一个起源的复杂的动力学在一个缩影的尺寸。数值方法的帮助下，路径跟踪软件，如approximately是必不可少的，以完成上述问题，特别是（a）。这些方法以一种协调的方式很好地理解复杂动力学的各个方面，并具有很大的潜力，可应用于其他领域。主要研究者西浦泰正于2002年获得日本数学会“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.