Oscillations and secondary bifurcations in nonlinear magnetoconvection
Oscillations and secondary bifurcations in nonlinear magnetoconvection
复制标题
非线性磁对流中的振荡和二次分岔
DOI：
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发表时间：
1993
期刊：
影响因子：
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通讯作者：
M. Proctor
中科院分区：
文献类型：
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作者：
A. Rucklidge;N. Weiss;D. P. Brownjohn;M. Proctor
由偏微分方程（PDE）控制的非线性系统中出现的复杂分岔结构可以通过研究适当的低阶振幅方程来解释。我们证明了这种方法的力量，考虑可压缩磁对流。数值实验揭示了从一个政权与亚临界Hopf分岔从静态解的过渡，一个有限振幅振荡持续存在，但没有从静态解的Hopf分岔。这种转变与一对零特征值的余维二分叉相关联。我们发现，分岔模式的偏微分方程确实是预测的二阶规范型方程（立方非线性）的Takens-Bogdanov分岔与Z2对称性。然后，我们通过添加五次非线性扩展这个方程，并分析所产生的系统。它的预测提供了一个定性准确的描述解决方案的完整偏微分方程在更广泛的参数值。用周期性的[O（2）]边界代替反射（Z2）侧边界条件，可以出现稳定的行波和调制波解;它们可以用三阶系统来描述。
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.