Complex turing patterns in non-linearly coupled systems

Recently pattern formation in layered structures, showing complicated superimposed patterns, has been modeled by coupling two Turing systems linearly, i.e., passively, such that the characteristic length scales of the independent systems are well separated. Here we propose a model of two non-linearly coupled Turing systems to study pattern formation in layered membrane-like structures, where the coupling plays an active role and changes the kinetics of the uncoupled systems. Extensive numerical simulations show that non-linear coupling generates a number of new regular patterns different from the ones observed earlier with linearly coupled systems. Some of them turn out to be superimposed patterns with different length scales, but many are not. Also, contrary to the linear coupling case, the strength of the non-linear coupling is found to play an important role in the formation and selection of patterns.