title: Instabilities and patterns in coupled reaction-diffusion layers.
abstract: We study instabilities and pattern formation in reaction-diffusion layers that are diffusively coupled. For two-layer systems of identical two-component reactions, we analyze the stability of homogeneous steady states by exploiting the block symmetric structure of the linear problem. There are eight possible primary bifurcation scenarios, including a Turing-Turing bifurcation that involves two disparate length scales whose ratio may be tuned via the interlayer coupling. For systems of n-component layers and nonidentical layers, the linear problem's block form allows approximate decomposition into lower-dimensional linear problems if the coupling is sufficiently weak. As an example, we apply these results to a two-layer Brusselator system. The competing length scales engineered within the linear problem are readily apparent in numerical simulations of the full system. Selecting a sqrt[2]:1 length-scale ratio produces an unusual steady square pattern.
authors.affiliation: Department of Mathematics, Wofford College Spartanburg, South Carolina 29303, USA. catllaaj@wofford.edu
authors.forename: Anne J
authors.initials: AJ
authors.lastname: Catllá
authors.forename: Amelia
authors.initials: A
authors.lastname: McNamara
authors.forename: Chad M
authors.initials: CM
authors.lastname: Topaz