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Bifurcations of periodic orbits with spatio-temporal symmetries

Rucklidge, A.M. and Silber, M. (1998) Bifurcations of periodic orbits with spatio-temporal symmetries. Nonlinearity, 11 (5). pp. 1435-1455. ISSN 1361-6544

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Abstract

Motivated by recent analytical and numerical work on two- and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic solutions of partial differential equations. Our approach is based on centre manifold reduction for maps, and is in the spirit of earlier work by Iooss (1986) on bifurcations of group orbits of spatially symmetric equilibria. Two examples, two-dimensional pulsating waves (PW) and three-dimensional alternating pulsating waves (APW), have discrete spatio-temporal symmetries characterized by the cyclic groups Z_n, n=2 (PW) and n=4 (APW). These symmetries force the Poincare' return map M to be the nth iterate of a map G: M=G^n. The group orbits of PW and APW are generated by translations in the horizontal directions and correspond to a circle and a two-torus, respectively. An instability of pulsating waves can lead to solutions that drift along the group orbit, while bifurcations with Floquet multiplier +1 of alternating pulsating waves do not lead to drifting solutions. The third example we consider, alternating rolls, has the spatio-temporal symmetry of alternating pulsating waves as well as being invariant under reflections in two vertical planes. This leads to the possibility of a doubling of the marginal Floquet multiplier and of bifurcation to two distinct types of drifting solutions. We conclude by proposing a systematic way of analysing steady-state bifurcations of periodic orbits with discrete spatio-temporal symmetries, based on applying the equivariant branching lemma to the irreducible representations of the spatio-temporal symmetry group of the periodic orbit, and on the normal form results of Lamb (1996). This general approach is relevant to other pattern formation problems, and contributes to our understanding of the transition from ordered to disordered behaviour in pattern-forming systems.

Item Type: Article
Copyright, Publisher and Additional Information: Copyright © 1998 IOP Publishing Ltd and LMS Publishing Ltd. This is an author produced version of an article published in Nonlinearity. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination. Movies relevant to this paper available at: http://www.maths.leeds.ac.uk/~alastair/papers/RS_STsymms_abs.html
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: 06 Feb 2006
Last Modified: 07 Jun 2014 16:03
Published Version: http://www.iop.org/EJ/abstract/0951-7715/11/5/015
Status: Published
Refereed: Yes
Identification Number: 10.1088/0951-7715/11/5/015
URI: http://eprints.whiterose.ac.uk/id/eprint/986

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