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October  2019, 12(6): 1669-1684. doi: 10.3934/dcdss.2019112

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

Sanjay Dharmavaram 1,, and  Timothy J. Healey 2,

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 $.

Keywords: Spherical harmonics, bifurcation theory, symmetry breaking.
Mathematics Subject Classification: Primary: 37G40, 58E09, 33C55; Secondary: 13A50.
Citation: Sanjay Dharmavaram, Timothy J. Healey. Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1669-1684. doi: 10.3934/dcdss.2019112
References:
[1]

F. H. Busse, Patterns of convection in spherical shells, J. Fluid Mech., 72 (1975), 67-85.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[6]

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Chelsea Publishing Company, New York, 1955.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. C. Matthews, Transcritical bifurcation with $ O(3) $ symmetry, Nonlinearity, 16 (2003), 1449-1471.  doi: 10.1088/0951-7715/16/4/315.  Google Scholar

[10]

B. Meyer, On the symmetries of spherical harmonics, Canad. J. Math., 6 (1954), 135-157.  doi: 10.4153/CJM-1954-016-2.  Google Scholar

[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.  Google Scholar

[12]

D. Sattinger, Group Theoretic Methods in Bifurcation Theory, Springer-Verlag, 1979.  Google Scholar

[13]

J. J. Sylvester, Note on spherical harmonics, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2 (1876), 291-307.   Google Scholar

[14]

E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

F. H. Busse, Patterns of convection in spherical shells, J. Fluid Mech., 72 (1975), 67-85.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Chelsea Publishing Company, New York, 1955.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. C. Matthews, Transcritical bifurcation with $ O(3) $ symmetry, Nonlinearity, 16 (2003), 1449-1471.  doi: 10.1088/0951-7715/16/4/315.  Google Scholar

[10]

B. Meyer, On the symmetries of spherical harmonics, Canad. J. Math., 6 (1954), 135-157.  doi: 10.4153/CJM-1954-016-2.  Google Scholar

[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.  Google Scholar

[12]

D. Sattinger, Group Theoretic Methods in Bifurcation Theory, Springer-Verlag, 1979.  Google Scholar

[13]

J. J. Sylvester, Note on spherical harmonics, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2 (1876), 291-307.   Google Scholar

[14]

E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959.  Google Scholar

[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.  Google Scholar

Figure 1.  Regular tetrahedron
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Figure 2.  A $\mathbb{T}$-invariant spherical harmonic for $\ell = 3$; (a) and (b) are diametrically opposite views
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Figure 3.  $\mathbb{O}\oplus Z_2^c$-invariant spherical harmonic for (a) $\ell = 4$ and (b) $\ell = 6$
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Figure 4.  (a) $\mathbb{O}$-invariant spherical harmonic for $\ell = 9$; (b) $\mathbb{O}^-$-invariant spherical harmonic for $\ell = 9$
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Figure 5.  $\mathbb{I}\oplus Z_2^c$-invariant basis functions for (a) $\ell = 6$ and (b) $\ell = 10$
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Figure 6.  $\mathbb{I}$-invariant spherical harmonic for $\ell = 15$
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Figure 7.  $D_{6}^d$-invariant spherical harmonic of order $\ell = 3$: (a) Front view and (b) top view
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Figure 8.  $D_4^d$-invariant spherical harmonic of order $\ell = 5$: (a) Front view and (b) top View
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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
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Figure 10.  The two basis functions (a) and (b) that span the subspace of $\mathbb{O}$-invariant spherical harmonics or order $\ell = 12$
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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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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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