Symmetry of Generic Bifurcations in Cubic Domains

Abstract

It is now well known that bifurcation problems arising from elliptic PDEs on finite domains may possess translational symmetries, even though translations cannot leave a finite domain invariant. These "hidden symmetries" are well understood when the domain is a multidimensional rectangle, a square, and a hemisphere. Hidden symmetries have two effects: they extend the symmetries of known solutions, and they make it possible to prove the existence of solutions that were not previously known. We determine the appropriate group action, which depends upon the mode numbers of the bifurcating solution. Throughout we consider only single modes (supported by absolutely irreducible representations). The group theory is considerably richer than for rectangular domains, because a cubic domain is invariant under the group Sn of all permutations of the coordinate axes. We specialize our results to the cases n = 1, 2, 3. When n = 2 we establish a conjecture of Crawford, made in the context of the Faraday experiment, by using hidden symmetries to predict the unexpected branches that he found. We also show that all such branches are pitchforks. The analysis of the case n = 3 is extensive and we find many new branches by using group elements that do not leave the appropriate fixed-point space invariant to define one-dimensional fixed-point spaces and applying the equivariant branching lemma. We illustrate sample planforms for these solutions when the mode numbers are (15, 6, 10), (42, 15, 35) and (21, 15, 35), the smallest numbers for which all pairs have a nontrivial common factor. Finally we observe that generic normal forms must vary wildly with the mode numbers, unlike the situation for rectangles. We briefly describe an algorithm to find normal forms in any specific case.