D. Gillis and M. Golubitsky
Patterns in square arrays of coupled cells
JMAA. 208 (1997) 487-509.
In recent years it has been observed that reaction-diffusion equations with Neumann boundary conditions (as well as other classes of PDE's) possess more symmetry than that which may be expected, and these "hidden" symmetries affect the generic types of bifurcation which occur (see Golubitsky et al.[6], Field et al. [5], Armbruster and Dangelmayr [1], Crawford et al. [2], Gomes and Stewart [8,9], and others). In addition, equilibria with more highly developed patterns may exist for these equations than might otherwise be expected. Epstein and Golubitsky [4] show that these symmetries also affect the discretizations of reaction-diffusion equations on an interval. In particular, equilibria of such systems may have well-defined patterns, which may be considered as a discrete analog of Turing patterns.
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.