Patterns in Square Arrays of Coupled Cells
JMAA 208 (1997) 487-509.
In this paper, we use an idea similar to the one in [4] to show that the same phenomena occurs in discretizations of reaction-diffusion equations on a square satisfying Neumann boundary conditions. Such discretizations lead to n x n square arrays of identically coupled cells. By embedding the original n x n array into a new 2n x 2n array, we can embed the Neumann boundary condition discretization in a periodic boundary condition discretization and increase the symmetry group of the equations from square symmetry to the symmetry group Gamma =D_4+Z_{2n}, which includes the discrete translation symmetries Z_{2n}^2.
This extra translation symmetry gives rise (generically) to branches of equilibria which restrict to the original n x n array yielding equilibria of the equations with Neumann boundary conditions. As shown in Figures 3-7 of Section 4, these equilibria may take the form of rolls or quilt-like solutions, which are comprised of square blocks of cells symmetrically arranged within the array, with each block containing its own internal symmetries. The notation used in the figures is explained in Section 3.
This paper is structured as follows. In Section 2 we discuss the embedding and the corresponding symmetry group Gamma. In Section 3 we list the irreducible representations of Gamma, and use the equivariant branching lemma to produce our solutions. The proofs of the results listed in this section are given in Section 6. The analysis of steady-state bifurcations on the large array of cells is a `discretized' version of the analysis in Dionne and Golubitsky [3] of bifurcations in planar reaction-diffusion equations satisfying periodic boundary conditions on a square lattice. In Section 4 we discuss the types of pattern that arise from these solutions. Finally, in Section 5 we show how to compute the eigenvalues of the Jacobian at a trivial equilibrium for a discretized system of differential equations.