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On the controllability of racing sailing boats with foils
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 |
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 $.
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. |
show all references
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. |







Group | Invariant Spherical Harmonic Basis | Order |
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where | ||
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Group | Invariant Spherical Harmonic Basis | Order |
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where | ||
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