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Hopf Bifurcation from Rotating Waves and Patterns in Physical Space

Gloubitsky, M., LeBlanc, V. G. and Melbourne, Ian (2000) Hopf Bifurcation from Rotating Waves and Patterns in Physical Space Journal of Nonlinear Science, 10 (1). pp. 69-101. ISSN 0938-8974

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Abstract

Hopf bifurcations from time periodic rotating waves to two frequency tori have been studied for a number of years by a variety of authors including Rand and Renardy. Rotating waves are solutions to partial differential equations where time evolution is the same as spatial rotation.Thus rotating waves can exist mathematically only in problems that have at least SO(2) symmetry. In this paper we study the effect on this Hopf bifurcation when the problem has more than SO(2) symmetry. These effects manifest themselves in physical space and not in phase space. We use as motivating examples the experiments of Gorman et al. on porous plug burner flames, of Swinney et al. on the Taylor-Couette system, and of a variety of people on meandering spiral waves in the Belousov-Zhabotinsky reaction. In our analysis we recover and complete Rand’s classification of modulated wavy vortices in the Taylor-Couette system.

It is both curious and intriguing that the spatial manifestations of the two frequency motions in each of these experiments is different, and it is these differences that we seek to explain. In particular, we give a mathematical explanation of the differences between the nonuniform rotation of cellular flames in Gorman’s experiments and the meandering of spiral waves in the Belousov-Zhabotinsky reaction.

Our approach is based on the center bundle construction of Krupa with compact group actions and its extension to noncompact group actions by Sandstede, Scheel, and Wulff.

Item Type: Article
Additional Information: This is a pre-copy-editing, author-prepared, peer-reviewed PDF of an article published in the Journal of Nonlinear Science, 10, 69-101. Click here to access the publisher's version. © 2000 Springer-Verlag New York.
Divisions: Faculty of Engineering and Physical Sciences > Mathematics
Depositing User: Mr Adam Field
Date Deposited: 27 May 2010 14:40
Last Modified: 23 Sep 2013 18:33
URI: http://epubs.surrey.ac.uk/id/eprint/1424

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