Abstract
In many diverse pattern-forming systems, the observed arrangements are square-symmetric or hexagonal. However, while many investigations have been devoted separately to the competition between these patterns and rolls, important questions still remain regarding their competition with one another. This paper demonstrates that both square and hexagonal patterns can be accounted for by the resonant interaction of two sets of modes with wave numbers in the ratio [Formula: see text], aligned to lie on a square lattice. The behavior of this system is investigated by expanding about the multiple bifurcation point where the stability of both sets of modes is marginal. The corresponding amplitude equations, constructed through simple symmetry arguments, are shown to encompass various forms of hexagonal and square-symmetric patterns within a single framework. This framework is then applied to the Swift–Hohenberg partial differential equation, where both hexagons and squares with up–down-asymmetry are found to occur subcritically.
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