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Oscillations and secondary bifurcations in nonlinear magnetoconvection

Rucklidge, A.M., Weiss, N.O., Brownjohn, D.P. and Proctor, M.R.E. (1993) Oscillations and secondary bifurcations in nonlinear magnetoconvection. Geophysical and Astrophysical Fluid Dynamics, 68 (1-4). pp. 133-150. ISSN 1029-0419


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Complicated bifurcation structures that appear in nonlinear systems governed by partial differential equations (PDEs) can be explained by studying appropriate low-order amplitude equations. We demonstrate the power of this approach by considering compressible magnetoconvection. Numerical experiments reveal a transition from a regime with a subcritical Hopf bifurcation from the static solution, to one where finite-amplitude oscillations persist although there is no Hopf bifurcation from the static solution. This transition is associated with a codimension-two bifurcation with a pair of zero eigenvalues. We show that the bifurcation pattern found for the PDEs is indeed predicted by the second-order normal form equation (with cubic nonlinearities) for a Takens-Bogdanov bifurcation with Z2 symmetry. We then extend this equation by adding quintic nonlinearities and analyse the resulting system. Its predictions provide a qualitatively accurate description of solutions of the full PDEs over a wider range of parameter values. Replacing the reflecting (Z2) lateral boundary conditions with periodic [O(2)] boundaries allows stable travelling wave and modulated wave solutions to appear; they could be described by a third-order system.

Item Type: Article
Copyright, Publisher and Additional Information: This is an author produced version of an article published in Geophysical and Astrophysical Fluid Dynamics, Taylor and Francis Ltd. This paper has been peer-reviewed but does not include the final publisher proof-corrections or journal pagination.
Keywords: magnetoconvection, bifurcation theory, amplitude equations, Takens-Bogdanov bifurcation, gluing bifurcation
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: 05 Jun 2014 17:05
Status: Published
Refereed: Yes
URI: http://eprints.whiterose.ac.uk/id/eprint/976

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