Helical Turing patterns in the Lengyel-Epstein model in thin cylindrical layers

Abstract

The formation of Turing patterns was investigated in thin cylindrical layers using the Lengyel-Epstein model of the chlorine dioxide-iodine-malonic acid reaction. The influence of the width of the layer W and the diameter D of the inner cylinder on the pattern with intrinsic wavelength l were determined in simulations with initial random noise perturbations to the uniform state for W < l/2 and D $ l or lower. We show that the geometric constraints of the reaction domain may result in the formation of helical Turing patterns with parameters that give stripes (b ¼ 0.2) or spots (b ¼ 0.37) in two dimensions. For b ¼ 0.2, the helices were composed of lamellae and defects were likely as the diameter of the cylinder increased. With b ¼ 0.37, the helices consisted of semi-cylinders and the orientation of stripes on the outer surface (and hence winding number) increased with increasing diameter until a new stripe appeared.

Introduction to Focus Issue: Oscillations and Dynamic Instabilities in Chemical Systems: Dedicated to Irving R. Epstein on occasion of his 70th birthday

“…18 (Two contributions on Turing patterns use this model in the issue. 19,20 ) Reversible complexation of an activator species to form an unreactive, immobile complex was shown to facilitate the formation of Turing patterns. 7 This idea is further elaborated in the work of T oth and Horv ath in this issue by immobilizing autocatalysts in ionic systems though selective binding, even for ions with equal mobilities.…”

“…34 In this issue, Tam as B ansagi and Annette Taylor investigate helical Turing patterns that form in cylindrical layers using the Lengyel-Epstein model. 19 The group of Oliver Steinbock tests the prediction that scroll waves can rotate around one-dimensional phase singularities, experimentally controlled by steps, troughs, and corners. 35 Stable, complex, 3D convection-driven fronts arising in the chloritetetrathionate reaction as a result of antagonistic thermal and solutal contribution to the density charge are reported by the group of Agota T oth.…”

Advanced Chemical Computing Using Discrete Turing Patterns in Arrays of Coupled Cells

“…A nonuniform concentration profile can thus be viewed as a discrete Turing pattern. In contrast, within a continuous tissue, the Turing instability leads to smooth spatiotemporal structures such as labyrinth (Rudovics et al, 1999;Asakura et al, 2011), dots (Ouyang et al, 1995;Rudovics et al, 1999;Vanag and Epstein, 2001), stripes (Ouyang et al, 1995), hexagons (Horvath et al, 2009;Asakura et al, 2011), or helical patterns in cylindrical layers (Bánsági and Taylor, 2015).…”

Pattern Propagation Driven by Surface Curvature

“…We conducted numerical simulations using the Brusselator [32] and Lengyel-Epstein (LE) models [33,34] on several curved surfaces with a parameter set indicating Turing instability for flat planes. We observed that the static pattern on a flat surface becomes a propagating pattern on several curved surfaces (see Fig.…”