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Convergence properties of the 8, 10 and 12 mode representations of quasipatterns

Rucklidge, A.M. and Rucklidge, W.J. (2003) Convergence properties of the 8, 10 and 12 mode representations of quasipatterns. Physica D: Nonlinear Phenomena, 178 (1-2). pp. 62-82. ISSN 0167-2789


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Spatial Fourier transforms of quasipatterns observed in Faraday wave experiments suggest that the patterns are well represented by the sum of 8, 10 or 12 Fourier modes with wavevectors equally spaced around a circle. This representation has been used many times as the starting point for standard perturbative methods of computing the weakly nonlinear dependence of the pattern amplitude on parameters. We show that nonlinear interactions of n such Fourier modes generate new modes with wavevectors that approach the original circle no faster than a constant times n^-2, and that there are combinations of modes that do achieve this limit. As in KAM theory, small divisors cause difficulties in the perturbation theory, and the convergence of the standard method is questionable in spite of the bound on the small divisors. We compute steady quasipattern solutions of the cubic Swift-Hohenberg equation up to 33rd order to illustrate the issues in some detail, and argue that the standard method does not converge sufficiently rapidly to be regarded as a reliable way of calculating properties of quasipatterns.

Item Type: Article
Copyright, Publisher and Additional Information: Copyright © 2003 Elsevier Science B.V. This is an author produced version of an article published in Physica D: Nonlinear Phenomena. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination.
Keywords: pattern formation, quasipatterns, Faraday waves, small divisors
Institution: The University of Leeds
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds)
Depositing User: A. M. Rucklidge
Date Deposited: 10 Feb 2006
Last Modified: 07 Jun 2014 18:23
Published Version: http://dx.doi.org/10.1016/S0167-2789(02)00792-3
Status: Published
Refereed: Yes
Identification Number: 10.1016/S0167-2789(02)00792-3
URI: http://eprints.whiterose.ac.uk/id/eprint/997

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