M. Golubitsky, I. Stewart and B. Dionne
Coupled Cells: Wreath Products and Direct Products
In: Dynamics, Bifurcation and Symmetry (P. Chossat, ed.)
NATO ARW Series, Kluwer, Amsterdam, 1994, 127-138.
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 depends
on properties of the coupling. We illustrate this point by showing how
the combination of "L" and "G" can lead to a symmetry group 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.