Parker, M. J., Gomes, M. G. M. and Stewart, Ian (2006) Forced symmetry-breaking of square lattice planforms. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS, 18 (1). pp. 223-255. doi:10.1007/s10884-005-9004-z

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Abstract

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 plan-forms 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-Benard convection. Subsequent studies will discuss other two- and three-dimensional lattices.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
Publisher:	SPRINGER
ISSN:	1040-7294
Official Date:	January 2006
Dates:	Date Event January 2006 UNSPECIFIED
Volume:	18
Number:	1
Number of Pages:	33
Page Range:	pp. 223-255
DOI:	10.1007/s10884-005-9004-z
Publication Status:	Published

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