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Spatial period-multiplying instabilities of hexagonal Faraday waves

Tse, D.P., Rucklidge, A.M., Hoyle, R.B. and Silber, M. (2000) Spatial period-multiplying instabilities of hexagonal Faraday waves. Physica D: Nonlinear Phenomena, 146 (1-4). pp. 367-387. ISSN 0167-2789


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A recent Faraday wave experiment with two-frequency forcing reports two types of `superlattice' patterns that display periodic spatial structures having two separate scales. These patterns both arise as secondary states once the primary hexagonal pattern becomes unstable. In one of these patterns (so-called `superlattice-II') the original hexagonal symmetry is broken in a subharmonic instability to form a striped pattern with a spatial scale increased by a factor of 2sqrt{3} from the original scale of the hexagons. In contrast, the time-averaged pattern is periodic on a hexagonal lattice with an intermediate spatial scale (sqrt{3} larger than the original scale) and apparently has 60 degree rotation symmetry. We present a symmetry-based approach to the analysis of this bifurcation. Taking as our starting point only the observed instantaneous symmetry of the superlattice-II pattern presented in and the subharmonic nature of the secondary instability, we show (a) that the superlattice-II pattern can bifurcate stably from standing hexagons; (b) that the pattern has a spatio-temporal symmetry not reported in [1]; and (c) that this spatio-temporal symmetry accounts for the intermediate spatial scale and hexagonal periodicity of the time-averaged pattern, but not for the apparent 60 degree rotation symmetry. The approach is based on general techniques that are readily applied to other secondary instabilities of symmetric patterns, and does not rely on the primary pattern having small amplitude.

Item Type: Article
Copyright, Publisher and Additional Information: Copyright © 2000 Elsevier Science Publishers 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: Faraday waves, secondary instabilities, spatial period-multiplying, superlattice patterns, averaged symmetries of attractors
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: 13 Jun 2014 20:31
Published Version: http://dx.doi.org/10.1016/S0167-2789(00)00124-X
Status: Published
Refereed: Yes
Identification Number: 10.1016/S0167-2789(00)00124-X
URI: http://eprints.whiterose.ac.uk/id/eprint/992

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