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 research hot spot in the field of pattern formation since it is one of the most effective methods of 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 researches on this issue have been conducted in the multilayered systems and their responses to the periodic forcing have not yet been well understood. In this work, the influences of the spatial periodic forcing on the Turing patterns in two linearly coupled layers described by the Brusselator (Bru) model and the Lengyel-Epstein (LE) model respectively have 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 as long as one of the external spatial forcing and the supercritical Turing mode (referred to as internal forcing mode) of the Bru layer is a 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 modes, the subcritical Turing mode in the LE layer cannot be excited. But the superlattice pattern can also be generated when the spatial resonance is satisfied. When the eigenmode of the LE layer is supercritical, a simple and robust hexagon pattern with its characteristic wavelength appears and responds 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.