J. Crawford, M. Golubitsky, M.G.M. Gomes, E. Knobloch and I. Stewart
Boundary Conditions as Symmetry Constraints
In: Singularity Theory and its Applications Warwick 1989, Part II
Lecture Notes in Mathematics, No. 1463
Springer-Verlag, Heidelberg (1991) 63-79
Fujii, Minmura, and Nishiura [1985] and Armbruster and Dangelmayr [1986, 1987] have observed that reaction-diffusion equations on the interval with Neumann boundary conditions can be viewed as restrictions of similar problems with periodic boundary conditions; and that this extension reveals the presence of additional symmetry constraints which affect the generic bifurcation phenomena. We show that more generally, similar observations hold for multi-dimensional rectangular domains with eigher Neumann or Dirichlet boundary conditions, and analyse the group-theoretic restrictions that this structure imposes upon bifurcations. We discuss a number of examples of these phenomena that arise in applications, including the Taylor-Couette experiment, Rayleigh-Benard convection, and the Faraday experiment.