Patterns in Square Arrays of Coupled Cells

Abstract

In recent years it has been observed that reaction]diffusion equations Ž . with Neumann boundary conditions as well as other classes of PDEs possess more symmetry than that which may be expected, and these ‘‘hidden’’ symmetries affect the generic types of bifurcation which occur Ž w x w x w x see Golubitsky et al. 6 , Field et al. 5 , Armbruster and Dangelmayr 1 , w x w x . Crawford et al. 2 , Gomes and Stewart 8, 9 , and others . In addition, equilibria with more highly developed patterns may exist for these equaw x tions 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. w x 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 = n square arrays of identically coupled cells. By embedding the original n = n array into a new 2n = 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