Bifurcation characteristics and spatial patterns in an integro-differential equation

Kiyoshi Toko, K. Yamafuji

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Spatial patterns are studied in reaction-diffusion systems where chemical reactions are coupled with diffusion. An integro-differential equation is obtained when the inhibitor has a much larger reaction rate than the activator. The periodic spatial pattern near the bifurcation point βc is described with a time-dependent Ginzburg-Landau equation derived using a perturbative method, which can be applied to the situation where the amplitude of the patterned solution develops gradually and also where it appears discontinuously at βc. Furthermore, a new type of potential is proposed to explain comprehensively the mechanisms of the bifurcation and the localized spatial pattern appearing very far from equilibrium. Since the potential is constructed from two internal variables, the bifurcation can be understood from the analogy with the first-order phase transition in equilibrium systems, and at the same time the appearance of patterns is shown to be analogous with the phase separation. The theoretical results are compared with numerical calculations.

Original languageEnglish
Pages (from-to)459-470
Number of pages12
JournalPhysica D: Nonlinear Phenomena
Volume44
Issue number3
DOIs
Publication statusPublished - Sep 1 1990

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Integrodifferential equations
Bifurcation (mathematics)
Spatial Pattern
Integro-differential Equation
differential equations
Bifurcation
Phase separation
Reaction rates
Chemical reactions
First-order Phase Transition
Ginzburg-Landau Equation
Phase transitions
Phase Separation
Bifurcation Point
Reaction Rate
Reaction-diffusion System
Chemical Reaction
Numerical Calculation
Inhibitor
Analogy

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Statistical and Nonlinear Physics

Cite this

Bifurcation characteristics and spatial patterns in an integro-differential equation. / Toko, Kiyoshi; Yamafuji, K.

In: Physica D: Nonlinear Phenomena, Vol. 44, No. 3, 01.09.1990, p. 459-470.

Research output: Contribution to journalArticle

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