P.C. Bressloff, J.D. Cowan, M. Golubitsky and P.J. Thomas

Scalar and pseudoscalar bifurcations motivated by pattern formation on the visual cortex

Nonlinearity. 14 (2001) 739-775.

Bosch Vivancos, Chossat, and Melbourne showed that two types of steady-state bifurcations are possible from trivial states when Euclidean equivariant systems are restricted to a planar lattice --- scalar and pseudo-scalar --- and began the study of pseudoscalar bifurcations. The scalar bifurcations have been well studied since they appear in planar reaction diffusion systems and in plane layer convection problems. Bressloff, Cowan, Golubitsky, Thomas, and Wiener showed that bifurcations in models of the visual cortex naturally contain both scalar and pseudoscalar bifurcations, due to a different action of the Euclidean group in that application.

In this paper, we review the symmetry discussion in Bressloff et al. and we continue the study of pseudoscalar bifurcations. Our analysis furthers the study of pseudoscalar bifurcations in three ways.

(1) We complete the classification of axial subgroups on the hexagonal lattice in the shortest wave vector case proving the existence of one new planform --- a solution with triangular D_3 symmetry.

(2) We derive bifurcation diagrams for generic bifurcations giving, in particular, the stability of solutions to perturbations in the hexagonal lattice. For the simplest (codimension zero) bifurcations, these bifurcation diagrams are identical to the ones derived by Golubitsky, Swift, and Knobloch in the case of Benard convection when there is a midplane reflection --- though the details in the analysis are more complicated.

(3) We discuss the types of secondary states that can appear in codimension one bifurcations (one parameter in addition to the bifurcation parameter), which includes time periodic states from rolls and hexagons solutions and drifting solutions from triangles (though the drifting solutions are always unstable near codimension one bifurcations). The essential difference between scalar and pseudeoscalar bifurcations appears in this discussion.