B. Dionne, M. Golubitsky and I. Stewart

Coupled cells with internal symmetry Part I: wreath products

*Nonlinearity.*
**9** (1996) 559-574.

In this paper and its sequel we study arrays of
coupled identical cells that possess a `global' symmetry group
*G*, and in which the cells posses their own `internal' symmetry
group *L*. We focus on general existence conditions for
symmetry-breaking steady-state and Hopf bifurcations. The global and
internal symmetries can combine in two quite different ways, depending
on how the internal symmetries affect the coupling. Algebraically, the symmetries either combine to give the wreath product of the two groups or
the direct product. Here we develop a theory for the wreath product:
we analyse the direct product case in Part II.

The wreath product case occurs when the coupling is invariant under
internal symmetries. The main objective of the paper is to relate the
patterns of steady-state and Hopf bifurcation that occur in systems with
the combined symmetry group *L* wreath *G* to the corresponding
bifurcations in systems with symmetry *L* or *G*. This
organizes the problem by reducing it to simpler questions whose answers
can often be read off from known results.

The basic existence theorem for steady-state bifurcation is the
equivariant branching lemma, which states that under appropriate
conditions there will be a symmetry-breaking branch of steady states for
any isotropy subgroup with a one-dimensional fixed-point subspace. We call
such an isotropy subgroup *axial*. The analogous result for
equivariant Hopf bifurcation involves isotropy subgroups with a
two-dimensional fixed-point subspace, which we call *C-axial*
because of an analogy involving a natural complex structure. Our main
results are classification theorems for axial and *C*-axial
subgroups in wreath products.

We study some typical examples, rings of cells in which the
internal symmetry group is *O(2)* and the global symmetry
group is dihedral. As these examples illustrate,
one striking consequence of our general results
is that systems with wreath product coupling
often have states in which some cells are performing nontrivial
dynamics, while others remain quiescent. We also discuss the
common occurrence of heteroclinic cycles in wreath product systems.