2(n)-splitting or edge-splitting? A manner of splitting in dissipative systems

Since early 90's, much attention has been paid to dynamic dissipative patte rns in laboratories, especially, self-replicating pattern (SRP) is one of t he most exotic phenomena. Employing model system such as the Gray-Scott mod el, it is confirmed also by numerics that SRP can be obtained via destabili zation of standing or traveling spots. SRP is a typical example of transien t dynamics, and hence it is not a priori clear that what kind of mathematic al framework is appropriate to describe the dynamics. A framework in this d irection is proposed by Nishiurar-Ueyama [16], i.e., hierarchy structure of saddle-node points, which gives a basis for rigorous analysis. One of the interesting observation is that when there occurs self-replication, then on ly spots (or pulses) located at the boundary (or edge) are able to split. I nternal ones do not duplicate at all. For ID-case, this means that the numb er of newly born pulses increases like 2k after k-th splitting, not 2(n)-sp litting where all pulses split simultaneously. The main objective in this a rticle is two-fold: One is to construct a local invariant manifold near the onset of self-replication, and derive the nonlinear ODE on it. The other i s to study the manner of splitting by analysing the resulting ODE, and answ er the question "2(n)-splitting or edge-splitting?" starting from a single pulse. It turns out that only the edge-splitting occurs, which seems a natu ral consequence from a physical point of view, because the pulses at edge a re easier to access fresh chemical resources than internal ones.