# American Institute of Mathematical Sciences

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

 1 Department of Mathematics, 1 Dent Dr, Bucknell University, Lewisburg, PA 17837, USA 2 Department of Mathematics, Cornell University, Ithaca, NY 14853, USA

* Corresponding author: Sanjay Dharmavaram

Received  May 2018 Revised  June 2018 Published  November 2018

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

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 $\mathbb{R}^3$.

Citation: Sanjay Dharmavaram, Timothy J. Healey. Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019112
##### References:
 [1] F. H. Busse, Patterns of convection in spherical shells, J. Fluid Mech., 72 (1975), 67-85. [2] P. Chossat, R. Lauterbach and I. Melbourne, Steady-state bifurcation with $O(3)$-symmetry, Archive for Rational Mechanics and Analysis, 113 (1991), 313-376. doi: 10.1007/BF00374697. [3] M. Golubitsky, D. Schaefer and I. Stewart, Singularities and Groups in Bifurcation Theory Volume II, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-4574-2. [4] T. J. Healey and H. Kielhöfer, Global Symmetry-Breaking Bifurcation for the van der WaalsCahnHilliard Model on the Sphere $S^2$, J Dyn Diff Equat, 27 (2015), 705-720. doi: 10.1007/s10884-013-9310-9. [5] T. J. Healey and S. Dharmavaram, Symmetry-breaking global bifurcation in a surface continuum phase-field model for lipid bilayer vesicles, SIAM J. Math. Anal., 49 (2017), 1027-1059. doi: 10.1137/15M1043716. [6] E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Chelsea Publishing Company, New York, 1955. [7] J. Hodgkinson, Harmonic functions with polyhedral symmetry, The Journal of London Mathematical Society, 10 (1935), 221-226. doi: 10.1112/jlms/s1-10.2.221. [8] G. H. Knightly and D. Sather, Buckled states of a spherical shell under uniform external pressure, Arch.Rat. Mech. Anal, 72 (1980), 315-380. doi: 10.1007/BF00248522. [9] P. C. Matthews, Transcritical bifurcation with $O(3)$ symmetry, Nonlinearity, 16 (2003), 1449-1471. doi: 10.1088/0951-7715/16/4/315. [10] B. Meyer, On the symmetries of spherical harmonics, Canad. J. Math., 6 (1954), 135-157. doi: 10.4153/CJM-1954-016-2. [11] E. G. C. Poole, Spherical harmonics having polyhedral symmetry, Proceedings of the London Mathematical Society, 33 (1932), 435-456. doi: 10.1112/plms/s2-33.1.435. [12] D. Sattinger, Group Theoretic Methods in Bifurcation Theory, Springer-Verlag, 1979. [13] J. J. Sylvester, Note on spherical harmonics, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2 (1876), 291-307. [14] E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959. [15] S. Zhao, T. J. Healey and Q. Li, Direct computation of two-phase icosahedral equilibria of lipid bilayer vesicles, Computer Meth. Appl. Mech. Engr., 314 (2017), 164-179. doi: 10.1016/j.cma.2016.07.011.

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##### References:
 [1] F. H. Busse, Patterns of convection in spherical shells, J. Fluid Mech., 72 (1975), 67-85. [2] P. Chossat, R. Lauterbach and I. Melbourne, Steady-state bifurcation with $O(3)$-symmetry, Archive for Rational Mechanics and Analysis, 113 (1991), 313-376. doi: 10.1007/BF00374697. [3] M. Golubitsky, D. Schaefer and I. Stewart, Singularities and Groups in Bifurcation Theory Volume II, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-4574-2. [4] T. J. Healey and H. Kielhöfer, Global Symmetry-Breaking Bifurcation for the van der WaalsCahnHilliard Model on the Sphere $S^2$, J Dyn Diff Equat, 27 (2015), 705-720. doi: 10.1007/s10884-013-9310-9. [5] T. J. Healey and S. Dharmavaram, Symmetry-breaking global bifurcation in a surface continuum phase-field model for lipid bilayer vesicles, SIAM J. Math. Anal., 49 (2017), 1027-1059. doi: 10.1137/15M1043716. [6] E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Chelsea Publishing Company, New York, 1955. [7] J. Hodgkinson, Harmonic functions with polyhedral symmetry, The Journal of London Mathematical Society, 10 (1935), 221-226. doi: 10.1112/jlms/s1-10.2.221. [8] G. H. Knightly and D. Sather, Buckled states of a spherical shell under uniform external pressure, Arch.Rat. Mech. Anal, 72 (1980), 315-380. doi: 10.1007/BF00248522. [9] P. C. Matthews, Transcritical bifurcation with $O(3)$ symmetry, Nonlinearity, 16 (2003), 1449-1471. doi: 10.1088/0951-7715/16/4/315. [10] B. Meyer, On the symmetries of spherical harmonics, Canad. J. Math., 6 (1954), 135-157. doi: 10.4153/CJM-1954-016-2. [11] E. G. C. Poole, Spherical harmonics having polyhedral symmetry, Proceedings of the London Mathematical Society, 33 (1932), 435-456. doi: 10.1112/plms/s2-33.1.435. [12] D. Sattinger, Group Theoretic Methods in Bifurcation Theory, Springer-Verlag, 1979. [13] J. J. Sylvester, Note on spherical harmonics, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2 (1876), 291-307. [14] E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959. [15] S. Zhao, T. J. Healey and Q. Li, Direct computation of two-phase icosahedral equilibria of lipid bilayer vesicles, Computer Meth. Appl. Mech. Engr., 314 (2017), 164-179. doi: 10.1016/j.cma.2016.07.011.
Regular tetrahedron
A $\mathbb{T}$-invariant spherical harmonic for $\ell = 3$; (a) and (b) are diametrically opposite views
$\mathbb{O}\oplus Z_2^c$-invariant spherical harmonic for (a) $\ell = 4$ and (b) $\ell = 6$
(a) $\mathbb{O}$-invariant spherical harmonic for $\ell = 9$; (b) $\mathbb{O}^-$-invariant spherical harmonic for $\ell = 9$
$\mathbb{I}\oplus Z_2^c$-invariant basis functions for (a) $\ell = 6$ and (b) $\ell = 10$
$\mathbb{I}$-invariant spherical harmonic for $\ell = 15$
$D_{6}^d$-invariant spherical harmonic of order $\ell = 3$: (a) Front view and (b) top view
$D_4^d$-invariant spherical harmonic of order $\ell = 5$: (a) Front view and (b) top View
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
The two basis functions (a) and (b) that span the subspace of $\mathbb{O}$-invariant spherical harmonics or order $\ell = 12$
Subgroups of $O(3)$ and their invariant spherical harmonic basis. Here $s\in\{0, 1\}, p, q\in\mathbb{N}\cup\{0\}$
 Group Invariant Spherical Harmonic Basis Order $\mathbb{T}$ $\mathcal{T}^s_6 \mathcal{T}^p_4 \mathcal{T}^q_3(1/r)\vert_{r=1}$ $6s+4p+3q$ $\mathbb{O}$ $\mathcal{O}_9^s \mathcal{O}^p_6 \mathcal{O}^q_4(1/r)\vert_{r=1}$ $9s+6p+4q$ $\mathbb{I}$ $\mathcal{I}_{15}^s\mathcal{I}^p_{10}\mathcal{I}^q_{6}(1/r)\vert_{r=1}$ $15s+10p+6q$ $\mathbb{T}\oplus Z_2^c$ $\mathcal{T}_6^s \mathcal{T}^{2p}_3 \mathcal{T}^q_4(1/r)\vert_{r=1}$ $6s+6p+4q$ $\mathbb{O}\oplus Z_2^c$ $\mathcal{O}^p_6 \mathcal{O}^q_4(1/r)\vert_{r=1}$ $6p+4q$ $\mathbb{I}\oplus Z_2^c$ $\mathcal{I}^p_{10}\mathcal{I}^q_{6}(1/r)\vert_{r=1}$ $10p+6q$ $\mathbb{O}^{-}$ $\mathcal{T}_4^p\mathcal{T}_3^q(1/r)\vert_{r=1}$ $4p+3q$ $Z_n$ $\hat{z}^p \mathcal{C}_{qn}(1/r)\vert_{r=1}$, $\hat{z}^p \mathcal{S}_{qn}(1/r)\vert_{r=1}$ $p+qn$ $D_n$ $\hat{z}^{2p} \mathcal{C}_{qn}(1/r)\vert_{r=1}$, $\hat{z}^{2p+1} \mathcal{S}_{qn}(1/r)\vert_{r=1}$ $2p+qn$, $2p+1+qn$ (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 $n$) $\hat{z}^{2p}\mathcal{C}_{qn}(1/r)\vert_{r=1}$ $2p+qn$
 Group Invariant Spherical Harmonic Basis Order $\mathbb{T}$ $\mathcal{T}^s_6 \mathcal{T}^p_4 \mathcal{T}^q_3(1/r)\vert_{r=1}$ $6s+4p+3q$ $\mathbb{O}$ $\mathcal{O}_9^s \mathcal{O}^p_6 \mathcal{O}^q_4(1/r)\vert_{r=1}$ $9s+6p+4q$ $\mathbb{I}$ $\mathcal{I}_{15}^s\mathcal{I}^p_{10}\mathcal{I}^q_{6}(1/r)\vert_{r=1}$ $15s+10p+6q$ $\mathbb{T}\oplus Z_2^c$ $\mathcal{T}_6^s \mathcal{T}^{2p}_3 \mathcal{T}^q_4(1/r)\vert_{r=1}$ $6s+6p+4q$ $\mathbb{O}\oplus Z_2^c$ $\mathcal{O}^p_6 \mathcal{O}^q_4(1/r)\vert_{r=1}$ $6p+4q$ $\mathbb{I}\oplus Z_2^c$ $\mathcal{I}^p_{10}\mathcal{I}^q_{6}(1/r)\vert_{r=1}$ $10p+6q$ $\mathbb{O}^{-}$ $\mathcal{T}_4^p\mathcal{T}_3^q(1/r)\vert_{r=1}$ $4p+3q$ $Z_n$ $\hat{z}^p \mathcal{C}_{qn}(1/r)\vert_{r=1}$, $\hat{z}^p \mathcal{S}_{qn}(1/r)\vert_{r=1}$ $p+qn$ $D_n$ $\hat{z}^{2p} \mathcal{C}_{qn}(1/r)\vert_{r=1}$, $\hat{z}^{2p+1} \mathcal{S}_{qn}(1/r)\vert_{r=1}$ $2p+qn$, $2p+1+qn$ (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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