Understanding of Spatio-temporal patterns by Singular Limit Methods

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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.