open access

https://doi.org/10.14943/83542

https://hdl.handle.net/2115/69146

A skeleton structure of self-replicating dynamics

英語

Bogdanov-Takens point

1997-10-01

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

396

1-27

A skeleton dynamics for the self-replicating patterns (SRP) of reaction diffusion system is presented. Self­replicating dynamics can be regarded as a transient process from a localized trigger to a stable Turing pattern or oscillatory Turing pattern. It looks like a reverse process of usual coarsening phenomena, i.e., the number of unit localized pattern increases until the domain is filled by them completely. SRP was found in several chemical reaction models, for instance, the Gray-Scott model as well as in real experi­ments. The most difficult point to describe SRP lies in the fact that it is truly a transient phenomenon in the sense that it can be captured neither as a definite object in dynamical system theory like an attractor nor an orbit itinerating among saddle points in the phase space. To our knowledge, it is not known that what kind of dynamical framework is suitable to clarify the behavior of SRP. The aim is to give a new point of view to describe such a transient dynamics of SRP on a finite interval. Especially we concen­trate on the basic mechanism causing SRP from a bifurcational view point by employing a new model system and its finite-dimensional con:.partment model which shares common qualitative features with the Gray-Scott model. By a careful anatomy of global bifurcation diagrams, the skeleton dynamics of SRP comes from a hierarchy structure of the subcritical bifurcating loops of oscillatory branches of pulse type. It should be noted that these loops themselves do not constitute the skeleton dynamics of SRP, but the ruins of them do it. In other words, the aftereffect of the hierarchy structure manifests the dynamics of SRP. The most important ingredient of an organizing center from which the whole hierarchy structure of SRP emerges is Bogdanov-Takens-Turing singularity as well as the existence of stable equilibrium point, which indicates universality of the above structure in the class of nonlinearities sharing this character.

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