Coupled Cells: Wreath Products and Direct Products

In this note we discuss the structure of systems of coupled cells (which we view as systems of ordinary differential equations) where symmetries of the system are obtained through the group G of global permutations of the cells and the group L of local internal symmetries of the dynamics in each cell. We show that even when the cells are assumed to be identical with identical coupling, the way that G and L combine to form the total symmetry group of the system Γ depends on properties of the coupling. We illustrate this point by showing how the combination of L with G can lead to a symmetry group Γ that is either a direct product or a wreath product. The symmetry group has strong implications for the dynamics of the system of cells, and the distinction between the two cases is substantial. This has important implications for the modeling of systems by coupled cells.