COUPLED CELLS WITH INTERNAL SYMMETRY .1. WREATH-PRODUCTS

In this paper and its sequel we study arrays of coupled identical cell s that possess a 'global' symmetry group g, and in which the cells pos sess their own 'internal' symmetry group L. We focus on general existe nce conditions for symmetry-breaking steady-state and Hopf bifurcation s. The global and internal symmetries can combine in two quite differe nt ways, depending on how the internal symmetries affect the coupling. Algebraically, the symmetries either combine to give the wreath produ ct L (sic) g of the two groups or the direct product L x g. Here we de velop a theory for the wreath product: we analyse the direct product c ase in the accompanying paper (henceforth referred to as II). The wrea th product case occurs when the coupling is invariant under internal s ymmetries. The main objective of the paper is to relate the patterns o f steady-state and Hopf bifurcation that occur in systems with the com bined symmetry group L (sic) g to the corresponding bifurcations in sy stems with symmetry L or g. This organizes the problem by reducing it to simpler questions whose answers can often be read off from known re sults. The basic existence theorem for steady-state bifurcation is the equivariant branching lemma, which states that under appropriate cond itions there will be a symmetry-breaking branch of steady states for a ny isotropy subgroup with a one-dimensional fixed-point subspace, We c all such an isotropy subgroup axial. The analogous result for equivari ant Hopf bifurcation involves isotropy subgroups with a two-dimensiona l fixed-point subspace, which we call C-axial because of an analogy in volving a natural complex structure. Our main results are classificati on theorems for axial and C-axial subgroups in wreath products. We stu dy some typical examples, rings of cells in which the internal symmetr y group is O(2) and the global symmetry group is dihedral. As these ex amples illustrate, one striking consequence of our general results is that systems with wreath product coupling often have states in which s ome cells are performing nontrivial dynamics, while others remain quie scent. We also discuss the common occurrence of heteroclinic cycles in wreath product systems.