Hopf bifurcations on cubic lattices

Group theoretic means are employed to analyse the Hopf bifurcation on pattern forming systems with the periodicity of the face-centred (FCC) and bodycentred (BCC) cubic lattices. We find all C-axial subgroups of the normal form symmetry group by first extending the symmetry to a larger group. There are 15 such solutions for the FCC lattice, of which at least 12 can be stable for appropriate parameter values. In addition, a number of subaxial solutions can bifurcate directly from the trivial solution, and quasiperiodic solutions can also exist. We find 33 C-axial solutions for the BCC lattice and their stability criteria. We discuss applications of the method of symmetry enlargement to other systems. A model-independent approach is taken throughout, and the results are applicable to a wide variety of pattern forming systems. This work is an extension of that done in Callahan T K (2000 Hopf bifurcations on the FCC lattice Proc. Int. Conf. on Differential Equations (Berlin, 1999) vol 1, ed Fiedler et al (Singapore: World Scientific) pp 154–6; 2003 Hopf bifurcations on cubic lattices Bifurcations, Symmetry and Patterns (Trends in Mathematics) ed J Buescu et al (Basel: Birkhauser) pp 123–7). Mathematics Subject Classification: 37G40, 35K57