Stability results for steady, spatially periodic planforms
Stability results for steady, spatially periodic planforms
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稳定的空间周期性平面形状的稳定性结果
DOI：
10.1088/0951-7715/10/2/002
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发表时间：
1995
期刊：
影响因子：
1.7
通讯作者：
A. Skeldon
中科院分区：
文献类型：
--
作者：
B. Dionne;M. Silber;A. Skeldon
等变分岔理论已被广泛用于研究E(2)-等变偏微分方程模拟的各种物理系统中通过对称破缺稳态分岔形成的模式。许多注意力集中在关于正方形或六边形晶格的双周期解上，对于这些解，分岔问题可以限制在有限维中心流形上。先前的研究分别对正方形和六边形晶格对称群使用了四维和六维表示，这反过来又允许确定正方形和卷或六边形和卷的相对稳定性。这里我们分别考虑正方形和六边形情况下的8维和12维不可约表示的可数无限集。这将先前的相对稳定性结果扩展到包括更多种类的分岔平台，并且也允许将滚动，正方形和六边形的稳定性建立到可数无限摄动集。在每一种情况下，我们推导了等变分支问题的泰勒展开式，并计算了这些解分支的线性轨道稳定性，这些解分支由等变分支引理保证存在。在这两种情况下，我们发现许多稳定性结果都是在泰勒展开中以三次阶建立的，尽管要完全确定某些状态的稳定性，需要高阶项。对于六边形晶格，由于泰勒展开中存在二次项，所有由等变分支引理保证的解分支一般都是不稳定的。为此，我们考虑了两种特殊情况：将二次项的系数设为零得到的简并分岔问题，以及存在额外反射对称时的分岔问题。
Equivariant bifurcation theory has been used extensively to study pattern formation via symmetry-breaking steady-state bifurcation in various physical systems modelled by E(2)-equivariant partial differential equations. Much attention has been focused on solutions that are doubly periodic with respect to a square or hexagonal lattice, for which the bifurcation problem can be restricted to a finite-dimensional centre manifold. Previous studies have used four- and six-dimensional representations for the square and hexagonal lattice symmetry groups respectively, which in turn allows the relative stability of squares and rolls or hexagons and rolls to be determined. Here we consider the countably infinite set of eight- and 12-dimensional irreducible representations for the square and hexagonal cases, respectively. This extends earlier relative stability results to include a greater variety of bifurcating planforms, and also allows the stability of rolls, squares and hexagons to be established to a countably infinite set of perturbations. In each case we derive the Taylor expansion of the equivariant bifurcation problem and compute the linear, orbital stability of those solution branches guaranteed to exist by the equivariant branching lemma. In both cases we find that many of the stability results are established at cubic order in the Taylor expansion, although to completely determine the stability of certain states, higher-order terms are required. For the hexagonal lattice, all of the solution branches guaranteed by the equivariant branching lemma are, generically, unstable due to the presence of a quadratic term in the Taylor expansion. For this reason we consider two special cases: the degenerate bifurcation problem that is obtained by setting the coefficient of the quadratic term to zero, and the bifurcation problem when an extra reflection symmetry is present.