A skeleton structure of self-replicating dynamics

Abstract

Self-replicating patterns (SRP) have been observed in several chemical reaction models, such as the Gray–Scott (GS) model, as well as in physical experiments. Watching these experiments (computational and physical) is like watching the more familiar coarsening processes but in reverse: the number of unit localized patterns increases until they fill the domain completely. Self-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 definition to characterize SRP. It cannot be described in terms of well-studied structures such as the attractor or a singular saddle orbit for a dynamical system. In this paper, we present a new point of view to describe the transient dynamics of SRP over a finite interval of time. We focus our attention on the basic mechanism causing SRP from a global bifurcational point of 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 points of folding bifurcation branches in parameter regions where the branches have ceased to exist. Thus, the skeleton structure mentioned in the title refers to the remains of bifurcation branches, the aftereffects of which are manifest in the dynamics of SRP. One of the natural and important problems is about the existence of an organizing center from which the whole hierarchical structure of limit points emerges. In our setting, the numerics suggests a strong candidate for that, i.e., Bogdanov–Takens–Turing (BTT) singularity together with the presence of a stable critical point, and so this indicates a universality of the above structure in the class of equations sharing this characteristic.