  We consider the symmetry-breaking steady state bifurcation of a
spatially-uniform equilibrium solution of E(2)-equivariant PDEs. We restrict
the space of solutions to those that are doubly-periodic with respect to a
square or hexagonal lattice, and consider the bifurcation problem restricted to
a finite-dimensional center manifold. For the square lattice we assume that the
kernel of the linear operator, at the bifurcation point, consists of 4 complex
Fourier modes, with wave vectors K_1=(a,b), K_2=(-b,a), K_3=(b,a), and
K_4=(-a,b), where a>b>0 are integers. For the hexagonal lattice, we assume that
the kernel of the linear operator consists of 6 complex Fourier modes, also
parameterized by an integer pair (a,b). We derive normal forms for the
bifurcation problems, which we use to compute the linear, orbital stability of
those solution branches guaranteed to exist by the equivariant branching lemma.
These solutions consist of rolls, squares, hexagons, a countable set of rhombs,
and a countable set of planforms that are superpositions of all of the Fourier
modes in the kernel. Since rolls and squares (hexagons) are common to all of
the bifurcation problems posed on square (hexagonal) lattices, this framework
can be used to determine their stability relative to a countable set of
perturbations by varying a and b. For the hexagonal lattice, we analyze the
degenerate bifurcation problem obtained by setting the coefficient of the
quadratic term to zero. The unfolding of the degenerate bifurcation problem
reveals a new class of secondary bifurcations on the hexagons and rhombs
solution branches.
