Transcritical bifurcation with O(3) symmetry
Transcritical bifurcation with O(3) symmetry
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具有 O(3) 对称性的跨临界分岔
DOI：
10.1088/0951-7715/16/4/315
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发表时间：
2003
期刊：
影响因子：
1.7
通讯作者：
P. Matthews
中科院分区：
文献类型：
--
作者：
P. Matthews
当球谐函数的次数l为偶数时，球对称分岔发生跨临界。在这种情况下，首阶分岔方程完全由对称性决定。给出了二维子空间中具有二面角对称性解的存在性和不存在性的几个新结果。对于大的l，存在交替排列的存在和不存在这样的解决方案。虽然所有的分支分支分支的固定解决方案是不稳定的，一个首选的解决方案是使用变分准则，该解决方案也只有一个正的特征值。它表明，轴对称状态是从来没有根据这个标准的首选解决方案。结果的存在性和稳定性的解决方案分支的偶数值l到l = 18，包括所有的解决方案在子空间的三维或更低。对于l = 6、10和12，优选的解具有二十面体对称性。
Bifurcation from spherical symmetry occurs transcritically when the degree l of the spherical harmonics is even. In this case the leading-order bifurcation equations are completely determined by the symmetry. Several new results are presented concerning the existence or non-existence of solutions with dihedral symmetry in two-dimensional subspaces. For large l, there is an alternating arrangement of existence and non-existence of such solutions. Although all bifurcating branches of stationary solutions are unstable, a preferred solution is identified using a variational criterion; this solution also has only one positive eigenvalue. It is shown that the axisymmetric state is never the preferred solution according to this criterion. Results on the existence and stability of solution branches are given for even values of l up to l = 18, including all solutions in subspaces of dimension three or lower. For l = 6, 10 and 12, the preferred solution has icosahedral symmetry.