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

  • * Corresponding author: Sanjay Dharmavaram

    * Corresponding author: Sanjay Dharmavaram 

The work of TJH was supported in part by the National Science Foundation through grant DMS-1613753, which is gratefully acknowledged

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  • 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∈\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 $.

    Mathematics Subject Classification: Primary: 37G40, 58E09, 33C55; Secondary: 13A50.


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  • Figure 1.  Regular tetrahedron

    Figure 2.  A $\mathbb{T}$-invariant spherical harmonic for $\ell = 3$; (a) and (b) are diametrically opposite views

    Figure 3.  $\mathbb{O}\oplus Z_2^c$-invariant spherical harmonic for (a) $\ell = 4$ and (b) $\ell = 6$

    Figure 4.  (a) $\mathbb{O}$-invariant spherical harmonic for $\ell = 9$; (b) $\mathbb{O}^-$-invariant spherical harmonic for $\ell = 9$

    Figure 5.  $\mathbb{I}\oplus Z_2^c$-invariant basis functions for (a) $\ell = 6$ and (b) $\ell = 10$

    Figure 6.  $\mathbb{I}$-invariant spherical harmonic for $\ell = 15$

    Figure 7.  $D_{6}^d$-invariant spherical harmonic of order $\ell = 3$: (a) Front view and (b) top view

    Figure 8.  $D_4^d$-invariant spherical harmonic of order $\ell = 5$: (a) Front view and (b) top View

    Figure 9.  One of the basis function that generate the two dimensional subspace of $\mathbb{D}_4\oplus Z_2^c$-invariant spherical harmonic of order $\ell = 4$: (a) Front view and (b) top view

    Figure 10.  The two basis functions (a) and (b) that span the subspace of $\mathbb{O}$-invariant spherical harmonics or order $\ell = 12$

    Table 1.  Subgroups of $O(3)$ and their invariant spherical harmonic basis. Here $s\in\{0, 1\}, p, q\in\mathbb{N}\cup\{0\}$

    GroupInvariant Spherical Harmonic BasisOrder
    $\mathbb{T}$ $\mathcal{T}^s_6 \mathcal{T}^p_4 \mathcal{T}^q_3(1/r)\vert_{r=1}$ $6s+4p+3q$
    $\mathbb{O}$ $\mathcal{O}_9^s \mathcal{O}^p_6 \mathcal{O}^q_4(1/r)\vert_{r=1}$ $9s+6p+4q$
    $\mathbb{I}$ $\mathcal{I}_{15}^s\mathcal{I}^p_{10}\mathcal{I}^q_{6}(1/r)\vert_{r=1}$ $15s+10p+6q$
    $\mathbb{T}\oplus Z_2^c$ $\mathcal{T}_6^s \mathcal{T}^{2p}_3 \mathcal{T}^q_4(1/r)\vert_{r=1}$ $6s+6p+4q$
    $\mathbb{O}\oplus Z_2^c$ $\mathcal{O}^p_6 \mathcal{O}^q_4(1/r)\vert_{r=1}$ $6p+4q$
    $\mathbb{I}\oplus Z_2^c$ $\mathcal{I}^p_{10}\mathcal{I}^q_{6}(1/r)\vert_{r=1}$ $10p+6q$
    $\mathbb{O}^{-}$ $\mathcal{T}_4^p\mathcal{T}_3^q(1/r)\vert_{r=1}$ $4p+3q$
    $Z_n$ $\hat{z}^p \mathcal{C}_{qn}(1/r)\vert_{r=1}$, $\hat{z}^p \mathcal{S}_{qn}(1/r)\vert_{r=1}$ $p+qn$
    $D_n$ $\hat{z}^{2p} \mathcal{C}_{qn}(1/r)\vert_{r=1}$, $\hat{z}^{2p+1} \mathcal{S}_{qn}(1/r)\vert_{r=1}$ $2p+qn$, $2p+1+qn$ (resp.)
    $D_n^z$ $\hat{z}^{p}\mathcal{C}_{qn}(1/r)\vert_{r=1}$ $p+qn$
    $Z_{2n}^-$ (even $n$), $Z_n\oplus Z_2^c$ (odd $n$) $\hat{z}^{2p+j} \mathcal{C}_{qn}(1/r)\vert_{r=1}$, $\hat{z}^{2p+j} \mathcal{S}_{qn}(1/r)\vert_{r=1}$, $2p+j+qn$
    where $j = qn (\text{ mod }2)$
    $Z_{2n}^-$ (odd $n$), $Z_n\oplus Z_2^c$ (even $n$) $\hat{z}^{2p} \mathcal{C}_{qn}(1/r)\vert_{r=1}$, $\hat{z}^{2p} \mathcal{S}_{qn}(1/r)\vert_{r=1}$ $2p+qn$
    $D_{2n}^d$ (even $n$), $D_n\oplus Z_2^c$ (odd $n$) $\hat{z}^{2p}\mathcal{C}_{2qn}(1/r)\vert_{r=1}$, $\hat{z}^{2p+1}\mathcal{S}_{(2q+1)n}(1/r)\vert_{r=1}$ $2p+2qn$, $2p+1+(2q+1)n$ (resp.)
    $D_{2n}^d$ (odd $n$), $D_n\oplus Z_2^c$ (even $n$) $\hat{z}^{2p}\mathcal{C}_{qn}(1/r)\vert_{r=1}$ $2p+qn$
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