Pattern formation in three-dimensional reaction-diffusion systems
type: Item
instance of: scholarly article
title: Pattern formation in three-dimensional reaction-diffusion systems
zbMATH Open document ID: 0935.35065
DOI: 10.1016/S0167-2789(99)00041-X
author: T. K. Callahan
author: Edgar Knobloch
published in: Physica D
publication date: 2000-04-24T00:00:00Z
review text: The work is based on a general theory that helps to derive normal forms of dynamical systems from a system of spatially isotropic three-dimensional coupled nonlinear partial differential equations of the reaction-diffusion type in the case when its zero solution loses stability against finite-wavelength perturbations (the Turing instability). Assuming periodic boundary conditions, the dynamical systems are generated by the truncation of the PDEs, projecting them onto a center manifold corresponding to a set of spatial harmonics based on three, four, or six different vectors. The projection is performed, using the corresponding representations of the isotropy group in the three-dimensional space. The three above-mentioned cases are referred to, respectively, as the simple cubic, face-centered cubic, and body-centered cubic (BCC) lattices. In the latter case, the derived normal form contains not only cubic, but also quadratic nonlinear terms. It is found that, in some cases, the consistent analysis demands higher-order (quintic) nonlinear terms to be added (which is not actually done in this work). In all the three cases, the normal forms of the dynamical systems depend on a single control parameter. For two particular reaction-diffusion models, viz., the Brusselator and Lengyel-Epstein ones, the control parameter is calculated as a function of actual parameters of the models. A chain of bifurcation diagrams is analyzed for each dynamical system, varying the control parameter. Different stationary solutions appearing in the diagrams are interpreted as solutions, with a certain spatial symmetry, to the underlying PDE system. In the dynamical system corresponding to the BCC lattice, all the stationary solutions are found to be unstable near the onset of the Turing instability; however, in a case when the coefficient in front of the quadratic nonlinear terms is small, that is relevant to the applications, secondary bifurcations occur close to the onset, stabilizing vario…
reviewed by: B. A. Malomed
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 92C15
Mathematics Subject Classification ID: 37L20
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 35Q80
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 37L10
zbMATH DE Number: 1370255
zbMATH Keywords: Turing instability
zbMATH Keywords: periodic boundary conditions
zbMATH Keywords: isotropy group
zbMATH Keywords: Brusselator
zbMATH Keywords: Turing patterns
zbMATH Keywords: bifurcation diagrams
zbMATH Keywords: secondary bifurcations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1370255