Equivariant bifurcation theory has been used to study pattern formation in various physical systems modelled by E(2)-equivariant partial differential equations. The existence of spatially doubly periodic solutions with respect to the square lattice has been the focus of much research. Previous studies have considered the four- and eight-dimensional representation of the square lattice, where the symmetry of the model is perfect. Here we consider the forced symmetry-breaking of the group orbits of translation free axial planforms in the four- and eight-dimensional representations. This problem is abstracted to the study of the action of the symmetry group of the perturbation on the group orbit of solutions. A partial classification for the behaviour of the group orbits is obtained, showing the existence of heteroclinic cycles and networks between equilibria. Possible areas of application are discussed including Faraday waves and Rayleigh–Bénard convection. Subsequent studies will discuss other two- and three-dimensional lattices.
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Parker, M.J., Gomes, M.G.M. & Stewart, I.N. Forced Symmetry-Breaking of Square Lattice Planforms. J Dyn Diff Equat 18, 223–255 (2006). https://doi.org/10.1007/s10884-005-9004-z
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DOI: https://doi.org/10.1007/s10884-005-9004-z