2n-Splitting or Edge-Splitting? A Manner of Splitting in Dissipative Systems

Shin Ichiro Ei, Yasumasa Nishiura, Kei Ichi Ueda

Research output: Contribution to journalArticlepeer-review

24 Citations (Scopus)


Since early 90's, much attention has been paid to dynamic dissipative patterns in laboratories, especially, self-replicating pattern (SRP) is one of the most exotic phenomena. Employing model system such as the Gray-Scott model, it is confirmed also by numerics that SRP can be obtained via destabilization of standing or traveling spots. SRP is a typical example of transient dynamics, and hence it is not a priori clear that what kind of mathematical framework is appropriate to describe the dynamics. A framework in this direction is proposed by Nishiura-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 only spots (or pulses) located at the boundary (or edge) are able to split. Internal ones do not duplicate at all. For 1D-case, this means that the number of newly born pulses increases like 2k after k-th splitting, not 2n-splitting where all pulses split simultaneously. The main objective in this article 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 is to study the manner of splitting by analysing the resulting ODE, and answer the question "2n-splitting or edge-splitting?" starting from a single pulse. It turns out that only the edge-splitting occurs, which seems a natural consequence from a physical point of view, because the pulses at edge are easier to access fresh chemical resources than internal ones.

Original languageEnglish
Pages (from-to)181-205
Number of pages25
JournalJapan Journal of Industrial and Applied Mathematics
Issue number2
Publication statusPublished - 2001 Jun
Externally publishedYes


  • Pulse solution
  • Reaction diffusion system
  • Saddle-node bifurcation
  • Self-replicating pattern
  • Turing pattern
  • Wave splitting

ASJC Scopus subject areas

  • Engineering(all)
  • Applied Mathematics


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