Coupled cells with internal symmetry: I. Wreath products

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 possess 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 L o G of the two groups or the direct product L × G. Here we develop a theory for the wreath product: we analyse the direct product case in the accompanying paper (henceforth referred to as 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 o 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. AMS classification scheme numbers: 20xx, 57T05 1. Internal and global symmetries Arrays of coupled oscillators have been studied by many authors [1, 2, 11]. It has been noted that when the oscillators are identical, symmetries are induced into the associated system of differential equations [13] and these symmetries depend on the exact pattern of coupling. For example, one popular configuration is a system of N cells coupled in a ring [2, 11]; this system has dihedral DN symmetry. Another popular pattern of coupling is all to all coupling where each cell is coupled to every other cell [16, 4]; this type of coupling of N cells induces SN permutation symmetry. We call symmetries induced by the pattern 0951-7715/96/020559+16$19.50 c © 1996 IOP Publishing Ltd and LMS Publishing Ltd 559