Coupled cells with internal symmetry: II. Direct products

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 of the internal group and the global group . 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 or separately. Unlike the wreath product case, the theory makes extensive use of the representation theory of compact Lie groups. Again the central algebraic task is to classify axial and -axial subgroups of the direct product and to relate them to axial and -axial subgroups of the two groups and . We demonstrate how the results lead to efficient classification by studying both steady state and Hopf bifurcation in rings of coupled cells, where and . In particular we show that for Hopf bifurcation the case n = 4 modulo 4 is exceptional, by exhibiting two extra types of solution that occur only for those values of n.