Oscillations and secondary bifurcations in nonlinear magnetoconvection
作者:
A.M. Rucklidge,
N.O. Weiss,
D.P. Brownjohn,
M.R. E. Proctor,
期刊:
Geophysical & Astrophysical Fluid Dynamics
(Taylor Available online 1993)
卷期:
Volume 68,
issue 1-4
页码: 133-150
ISSN:0309-1929
年代: 1993
DOI:10.1080/03091929308203565
出版商: Taylor & Francis Group
关键词: Magnetoconvection;bifurcation theory;amplitude equations;Takens—Bogdanov bifurcation;gluing bifurcation
数据来源: Taylor
摘要:
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 Z2symmetry. 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.
点击下载:
PDF (923KB)
返 回