Stability Results for Steady, Spatially–Periodic Planforms

Equivariant bifurcation theory has been used extensively to study pattern formation via symmetry–breaking steady state bifurcation in various physical systems modeled by E(2)– equivariant partial differential equations. Much attention has focussed on solutions that are doubly–periodic with respect to a square or hexagonal lattice, for which the bifurcation problem can be restricted to a finite–dimensional center manifold. Previous studies have used four– and six–dimensional representations for the square and hexagonal lattice symmetry groups respectively, which in turn allows the relative stability of squares and rolls or hexagons and roll to be determined. Here we consider the countably infinite set of eight– and twelve– dimensional irreducible representations for the square and hexagonal cases, respectively. This extends earlier relative stability results to include a greater variety of bifurcating planforms, and also allows the stability of rolls, squares and hexagons to be established to a countably infinite set of perturbations. In each case we derive the Taylor expansion of the equivariant bifurcation problem and compute the linear, orbital stability of those solution branches guaranteed to exist by the equivariant branching lemma. In both cases we find that many of the stability results are established at cubic order in the Taylor expansion, although to completely determine the stability of certain states, higher order terms are required. For the hexagonal lattice, all of the solution branches guaranteed by the equivariant branching lemma are, generically, unstable due to the presence of a quadratic term in the Taylor expansion. For this reason we consider two special cases: the degenerate bifurcation problem that is obtained by setting the coefficient of the quadratic term to zero, and the bifurcation problem when an extra reflection symmetry is present.