COUPLED CELLS WITH INTERNAL SYMMETRY .2. DIRECT PRODUCTS

We continue the study of arrays of coupled identical cells that posses s both global and internal symmetries, begun in part I. Here we concen trate on the 'direct product' case, for which the symmetry group of th e system decomposes as the direct product L x g of the internal group L and the global group g. Again, the main aim is to find general exist ence conditions for symmetry-breaking steady-state and Hopf bifurcatio ns by reducing the problem to known results for systems with symmetry L or g separately. Unlike the wreath product case, the theory makes ex tensive use of the representation theory of compact Lie groups. 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 subgroup s of the two groups L and g. We demonstrate how the results lead to ef ficient classification by studying both steady state and Hopf bifurcat ion in rings of coupled cells, where L = O(2) and g = D-n. In particul ar we show that for Hopf bifurcation the case n = 4 module 4 is except ional, by exhibiting two extra types of solution that occur only for t hose values of n.