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Rucklidge, AM and Rucklidge, WJ (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
Abstract
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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2003 Elsevier Science B.V. This is an author produced version of a paper published in Physica D: Nonlinear Phenomena. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Pattern formation; Quasipatterns; Faraday waves; Small divisors |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | A. M. Rucklidge |
Date Deposited: | 14 Jul 2016 15:11 |
Last Modified: | 01 Feb 2018 11:27 |
Published Version: | http://dx.doi.org/10.1016/S0167-2789(02)00792-3 |
Status: | Published |
Publisher: | Elsevier |
Refereed: | Yes |
Identification Number: | 10.1016/S0167-2789(02)00792-3 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:74786 |
Available Versions of this Item
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Convergence properties of the 8, 10 and 12 mode representations of quasipatterns. (deposited 10 Feb 2006)
- Convergence properties of the 8, 10 and 12 mode representations of quasipatterns. (deposited 14 Jul 2016 15:11) [Currently Displayed]