Dharmavaram, Sanjay; Healey, Timothy J. Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry. (English) Zbl 1422.37032 Discrete Contin. Dyn. Syst., Ser. S 12, No. 6, 1669-1684 (2019). Summary: We consider bifurcation problems in the presence of \( O(3) \) symmetry. Well known group-theoretic techniques enable the classification of all maximal isotropy subgroups of \( O(3) \), with associated mode numbers \(\ell \in \mathbb{N} \), leading to 1-dimensional fixed-point subspaces of the \( (2\ell+1) \)-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 \( 2\ell+1 \) 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 \( \mathbb{R}^3 \). MSC: 37G40 Dynamical aspects of symmetries, equivariant bifurcation theory 58E09 Group-invariant bifurcation theory in infinite-dimensional spaces 33C55 Spherical harmonics 13A50 Actions of groups on commutative rings; invariant theory Keywords:spherical harmonics; bifurcation theory; symmetry breaking PDFBibTeX XMLCite \textit{S. Dharmavaram} and \textit{T. J. Healey}, Discrete Contin. Dyn. Syst., Ser. S 12, No. 6, 1669--1684 (2019; Zbl 1422.37032) Full Text: DOI