Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry

Abstract

We consider bifurcation problems in the presence of \begin{document}$ O(3) $\end{document} symmetry. Well known group-theoretic techniques enable the classification of all maximal isotropy subgroups of \begin{document}$ O(3) $\end{document} , with associated mode numbers \begin{document}$\ell∈\mathbb{N} $\end{document} , leading to 1-dimensional fixed-point subspaces of the \begin{document}$ (2\ell+1) $\end{document} -dimensional space of spherical harmonics. In each case the so-called equivariant branching lemma can then be used to establish the existence of a local branch of bifurcating solutions having the symmetry of the respective subgroup. To first-order, such a branch is a precise linear combination of the \begin{document}$ 2\ell+1 $\end{document} spherical harmonics, which we call the bifurcation direction. Our work here is focused on the direct construction of these bifurcation directions, complementing the above-mentioned classification. The approach is an application of a general method for constructing families of symmetric spherical harmonics, based on differentiating the fundamental solution of Laplace's equation in \begin{document}$ \mathbb{R}^3 $\end{document} .