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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Abstract
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 |
| 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: | 08 Feb 2013 16:48 |
| Status: | Published |
| Refereed: | Yes |
| URI: | http://eprints.whiterose.ac.uk/id/eprint/976 |
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