A skeleton structure of self-replicating dynamics

Self-replicating patterns (SRP) have been observed in several chemical reac tion models, such as the Gray-Scott (GS) model, as well as in physical expe riments, Watching these experiments (computational and physical) is like wa tching the more familiar coarsening processes but in reverse: the number of unit localized patterns increases until they fill the domain completely. S elf-replicating dynamics, then, can be regarded as a transient process from a localized trigger to a stable stationery or oscillating Turing pattern, Since it is a transient process, it is very difficult to give a suitable de finition to characterize SRP, It cannot be described in terms of well-studi ed structures such as the attractor or a singular saddle orbit for a dynami cal system. In this paper, we present a new point of view to describe the t ransient dynamics of SRP over a finite interval of time. We focus our atten tion on the basic mechanism causing SRP from a global bifurcational point o f view and take our clues from two model systems including the GS model. A careful analysis of the anatomy of the global bifurcation diagram suggests that the dynamics of SRP is related to a hierarchical structure of limit po ints of folding bifurcation branches in parameter regions where the branche s have ceased to exist. Thus, the skeleton structure mentioned in the title refers to the remains of bifurcation branches, the aftereffects of which a re manifest in the dynamics of SRP. One of the natural and important proble ms is about the existence of an organizing center from which the whole hier archical structure of limit points emerges. In our setting, the numerics su ggests a strong candidate for that, i.e., Bogdanov-Takens-Turing (BTT) sing ularity together with the presence of a stable critical point, and so this indicates a universality of the above structure in the class of equations s haring this characteristic. (C) 1999 Elsevier Science B.V. All rights reser ved.