Quasipatterns versus superlattices resulting from the superposition of two hexagonal patterns

Stéphan Fauve; Gérard Iooss

doi:10.1016/j.crme.2019.03.006

Patterns and dynamics: homage to Pierre Coullet / Formes et dynamique : hommage à Pierre Coullet

Quasipatterns versus superlattices resulting from the superposition of two hexagonal patterns

Stéphan Fauve ¹ ; Gérard Iooss ²

¹ Laboratoire de physique statistique, École normale supérieure, PSL Research University, UPMC Université Paris-6, Sorbonne Universités, Université Paris-Diderot, Sorbonne Paris-Cité, CNRS, 24, rue Lhomond, 75005 Paris, France
² Université Côte d'Azur, CNRS, LJAD, Parc Valrose 06108, Nice cedex 2, France

Comptes Rendus. Mécanique, Patterns and dynamics: homage to Pierre Coullet / Formes et dynamique: hommage à Pierre Coullet, Volume 347 (2019) no. 4, pp. 294-304.

Abstract
Résumé

We present a short review of the experimental observations and mechanisms related to the generation of quasipatterns and superlattices by the Faraday instability with two-frequency forcing. We show how two-frequency forcing makes possible triad interactions that generate hexagonal patterns, twelvefold quasipatterns or superlattices that consist of two hexagonal patterns rotated by an angle α relative to each other. We then consider which patterns could be observed when α does not belong to the set of prescribed values that give rise to periodic superlattices. Using the Swift–Hohenberg equation as a model, we find that quasipattern solutions exist for nearly all values of α. However, these quasipatterns have not been observed in experiments with the Faraday instability for $α \neq π / 6$ . We discuss possible reasons and mention a simpler framework that could give some hint about this problem.

Nous présentons une courte revue des observations expérimentales et des mécanismes qui ont permis d'engendrer des structures quasi cristallines à l'aide de l'instabilité de Faraday sous l'effet d'une excitation périodique comportant deux fréquences. Nous montrons comment l'excitation à deux fréquences permet d'obtenir des triades de vecteurs d'onde résonantes qui induisent la formation de structures hexagonales, de structures quasi cristallines dodécagonales ou de super-réseaux résultant de la superposition de deux réseaux d'hexagones tournés l'un par rapport à l'autre d'un angle α. Nous considérons ensuite quelles sont les structures obtenues lorsque α ne prend pas la série de valeurs discrètes conduisant à un super-réseau périodique. Nous montrons sur l'équation de Swift–Hohenberg qu'il existe dans ce cas des solutions à symétrie quasi cristalline pour presque toutes les valeurs de α. Nous discutons les raisons possibles pour lesquelles ces solutions n'ont pas été observées expérimentalement pour $α \neq π / 6$ et nous mentionnons un cadre plus simple qui permettrait d'éclaircir cette question.

Received: 2018-08-28
Accepted: 2018-11-09
Published online: 2019-04-01

DOI: 10.1016/j.crme.2019.03.006

Keywords: Bifurcations, Asymptotic series, Quasicrystals, Instability in fluids, Small divisors
Mots-clés : Bifurcations, Séries asymptotiques, Quasi-cristaux, Instabilité dans les fluides, Petits diviseurs

Author's affiliations:

Stéphan Fauve ¹; Gérard Iooss ²

¹ Laboratoire de physique statistique, École normale supérieure, PSL Research University, UPMC Université Paris-6, Sorbonne Universités, Université Paris-Diderot, Sorbonne Paris-Cité, CNRS, 24, rue Lhomond, 75005 Paris, France
² Université Côte d'Azur, CNRS, LJAD, Parc Valrose 06108, Nice cedex 2, France

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     author = {St\'ephan Fauve and G\'erard Iooss},
     title = {Quasipatterns versus superlattices resulting from the superposition of two hexagonal patterns},
     journal = {Comptes Rendus. M\'ecanique},
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Stéphan Fauve; Gérard Iooss. Quasipatterns versus superlattices resulting from the superposition of two hexagonal patterns. Comptes Rendus. Mécanique, Patterns and dynamics: homage to Pierre Coullet / Formes et dynamique: hommage à Pierre Coullet, Volume 347 (2019) no. 4, pp. 294-304. doi : 10.1016/j.crme.2019.03.006. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2019.03.006/

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