B. Dionne, M. Golubitsky and I. Stewart
Coupled Cells with Internal
Symmetry Part II: Direct Products
Nonlinearity 9 (1996) 575-599
We continue the study of arrays of coupled identical cells that possess
both global and internal symmetries, begun in Part I. Here we
concentrate on the `direct product' case, for which the symmetry group
of the system decomposes as the direct product LXG
of the internal group L and the global group G. Again,
the main aim is to find general existence conditions for symmetry-
breaking steady-state and Hopf bifurcations by reducing the problem to
known results for systems with symmetry L or G separately.
Unlike the wreath product case, the theory makes extensive use of the
representation theory of compact Lie groups, and we have provided an
appendix to summarise the main ideas required. Again the central
algebraic task is to classify axial and C-axial subgroups of the
direct product and to relate them to axial and C-axial subgroups
of the two groups L and G. We demonstrate how the
results lead to efficient classification by studying both steady state
and Hopf bifurcation in rings of coupled cells, where L = O(2) and
G = D_n. In particular we show that for Hopf bfiurcation the case
n = 0 modulo 4 is exceptional, by exhibiting two extra types of
solution that occur only for those values of n.