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Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry. (English) Zbl 1422.37032

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
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