Coupled Cells with Internal Symmetry Part I: Wreath Products
Nonlinearity 9 (1996) 559-574.
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.