Hopf bifurcation with cubic symmetry and instability of ABC

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Hopf bifurcation with cubic symmetry and instability of ABC

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dc.contributor.author Ashwin, Peter en_GB
dc.contributor.author Podvigina, Olga en_GB
dc.contributor.department University of Exeter; Observatoire de la Côte d’Azur, Nice; International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Moscow; Lomonosov Moscow State University en_GB
dc.date.accessioned 2008-03-14T10:48:01Z en_GB
dc.date.accessioned 2011-01-25T10:33:51Z en_US
dc.date.accessioned 2013-03-20T12:32:54Z
dc.date.issued 2003 en_GB
dc.description.abstract We examine the dynamics of generic Hopf bifurcation in a system that is symmetric under the action of the rotational symmetries of the cube. We classify the generic branches of periodic solutions at bifurcation; there are generically 27 branches corresponding to maximal symmetries, organized into five symmetry types. There are also up to 22 periodic solution branches of two other symmetry types. These results are found by examination of the normal form (with S1 normal-form symmetry) for the bifurcation truncated at the third order. In addition to the periodic branches whose branching and stability we find, there are several branches of tori, homoclinic bifurcations and chaotic attractors in the dynamics of the third-order normal form. Since many of these features are not amenable to analysis, we give some numerical examples. On breaking the normal form symmetry, there may be breakup of the branches of tori, but the predictions for the periodic solutions will be reliable. For the Navier-Stokes equations with a particular forcing, an ABC flow is a dynamically stable solution for small Reynolds numbers R. For the most symmetric case, A = B = C = 1, the first instability of this system is a Hopf bifurcation at R = 13.04 with rotational symmetry of the cube. We use our normal-form analysis to explain the observed behaviour of solutions at this primary instability. Numerical simulations show that there is supercritical branching to rotating waves that alternate between the three axes, which undergo secondary Hopf bifurcation to a 2-torus at approximately R = 13.09. The eight symmetrically related tori break up and then merge to form a chaotic attractor with full symmetry. We can explain all these features by use of the generic third-order normal form and S1 normal-form symmetry-breaking terms. en_GB
dc.identifier.citation 459 (2035), pp. 1801-1827 en_GB
dc.identifier.doi 10.1098/rspa.2002.1090 en_GB
dc.identifier.uri http://hdl.handle.net/10036/20712 en_GB
dc.language.iso en en_GB
dc.publisher Royal Society en_GB
dc.relation.url http://dx.doi.org/10.1098/rspa.2002.1090 en_GB
dc.relation.url http://publishing.royalsociety.org/index.cfm?page=1086# en_GB
dc.subject fluid instability en_GB
dc.subject hopf bifurcation en_GB
dc.subject ABC flow en_GB
dc.subject symmetry en_GB
dc.title Hopf bifurcation with cubic symmetry and instability of ABC en_GB
dc.type Article en_GB
dc.type Preprint en_GB
dc.date.available 2008-03-14T10:48:01Z en_GB
dc.date.available 2011-01-25T10:33:51Z en_US
dc.date.available 2013-03-20T12:32:54Z
dc.identifier.issn 1364-5021 en_GB
dc.identifier.issn 1471-2946 en_GB
dc.description Copyright © 2003 The Royal Society. NOTICE: This is the author’s version of a work accepted for publication by The Royal Society. The definitive version was subsequently published in Proceedings of the Royal Society A, Vol 459, Number 2035, online 28 May 2003 and in print 8 July 2003, DOI:10.1098/rspa.2002.1090 en_GB
dc.identifier.journal Proceedings of The Royal Society A en_GB


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