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Design of Parametrically Forced Patterns and Quasipatterns

Rucklidge, AM and Silber, M (2009) Design of Parametrically Forced Patterns and Quasipatterns. SIAM Journal on Applied Dynamical Systems, 8 (1). 298 - 347 (50). ISSN 1536-0040

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Abstract

The Faraday wave experiment is a classic example of a system driven by parametric forcing, and it produces a wide range of complex patterns, including superlattice patterns and quasipatterns. Nonlinear three-wave interactions between driven and weakly damped modes play a key role in determining which patterns are favored. We use this idea to design single and multifrequency forcing functions that produce examples of superlattice patterns and quasipatterns in a new model PDE with parametric forcing. We make quantitative comparisons between the predicted patterns and the solutions of the PDE. Unexpectedly, the agreement is good only for parameter values very close to onset. The reason that the range of validity is limited is that the theory requires strong damping of all modes apart from the driven pattern-forming modes. This is in conflict with the requirement for weak damping if three-wave coupling is to influence pattern selection effectively. We distinguish the two different ways that three-wave interactions can be used to stabilize quasipatterns, and we present examples of 12-, 14-, and 20-fold approximate quasipatterns. We identify which computational domains provide the most accurate approximations to 12-fold quasipatterns and systematically investigate the Fourier spectra of the most accurate approximations.

Item Type: Article
Copyright, Publisher and Additional Information: © 2009 SIAM publications. Reproduced in accordance with the publisher's self-archiving policy.
Keywords: pattern formation, quasipatterns, superlattice patterns, mode interactions, Faraday waves, quasi-periodic patterns, driven surface-waves, superlattice patterns, hexagonal patterns, crystals, instability, systems, symmetries, resonances
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: Symplectic Publications
Date Deposited: 12 Oct 2011 12:15
Last Modified: 15 Sep 2014 03:52
Published Version: http://dx.doi.org/10.1137/080719066
Status: Published
Publisher: SIAM Publications
Identification Number: 10.1137/080719066
URI: http://eprints.whiterose.ac.uk/id/eprint/43326

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