Effects of spatial periodic forcing on Turing patterns in two-layer coupled reaction diffusion system

Abstract

Periodic forcing of pattern-forming systems is always a hot point in the field of pattern formation since it is one of the most effective methods for controlling patterns. In reality, most of the pattern-forming systems are the multilayered systems, in which each layer is a reaction-diffusion system coupled to adjacent layers. However, few works on this issue has been done in the multilayered systems and their response to the periodic forcing has yet been well understood. In this paper, the influence of the spatial periodic forcing on the Turing patterns in a two linearly coupled layers described by the Brusselator（Bru）model and the Lengyel-Epstein（LE）model respectively has been investigated by introducing a spatial periodic forcing into the LE layer. It is found that the subcritical Turing mode in the LE layer can be excited either the spatial forcing or the supercritical Turing mode (refers to as internal forcing mode) in the Bru layer is served as the longer wave mode. These three modes interact together and give rise to complex patterns with three different spatial scales. If both the spatial forcing mode and the internal forcing mode are the short wave mode, the subcritical Turing mode in the LE layer cannot be excited. Only the spatial forcing mode and the internal forcing mode interact each other in this case, giving rise to superlattice patterns when they are satisfied the spatial resonance condition. When the eigenmode of the LE layer is supercritical, a simple and robust hexagon pattern with its characteristic wavelength appears and response to the spatial forcing only when the forcing intensity is large enough. It is found that the wave number of forcing has a powerful influence on the spatial symmetry of patterns.

Effects of square spatial periodic forcing on oscillatory hexagon patterns in coupled reaction–diffusion systems

“…10,[31][32][33][34][35][36][37] Recent experiments and numerical simulations have shown that coupled systems are susceptible to external forcing. [38][39][40][41][42][43] Several studies have been devoted to external forces that are uniform in space but periodic in time. For instance, Mı ´guez et al 37 experimentally investigated the interaction of Turing patterns in a diffusioncoupled reaction diffusion system by illuminating each of the layers.…”

“…10,31–37 Recent experiments and numerical simulations have shown that coupled systems are susceptible to external forcing. 38–43 Several studies have been devoted to external forces that are uniform in space but periodic in time. For instance, Míguez et al .…”