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References
Aronson, D. G., Golubitsky, M. & Krupa, M. [1990]. Coupled arrays of Josephson Junctions and bifurcation of maps with S N symmetry, submitted to Nonlinearity.
Babskii, V. G. & Sklovskaia, I. L. [1971]. On convection in a self-gravitating fluid sphere with internal heating. Prikl. Mat. Mekh. 35, 951–965.
Baldwin, P. [1967]. Finite amplitude convection in a self-gravitating fluid sphere containing heat sources. Proc. Camb. Phil. Soc. 63, 855–870.
Bartsch, T. [1988]. Critical orbits of invariant functionals and symmetry breaking. Preprint, Universität Heidelberg.
Bredon, G. E. [1972]. Introduction to Compact Transformation Groups. Academic Press, New York.
Bröcker, T. & Tom Dieck, T. [1985]. Representations of compact Lie groups. Springer-Verlag, Heidelberg-Berlin-New York.
Busse, F. H. [1975]. Pattern of convection in spherical shells. J. Fluid Mech. 72, 65–85.
Busse, F. H. & Riahi, N. [1982]. Pattern of convection in spherical shells, II. J. Fluid Mech. 123, 283–301.
Busse, F. H. & Riahi, N. [1988]. Mixed-mode patterns of bifurcation from spherically symmetric basic states. Nonlinearity 1, 379–388.
Chandrasekhar, S. [1961, 1970]. Hydrodynamic and hydromagnetic stability. Dover, New York.
Childress, S. [1977]. Convective dynamos, in: Problems of stellar convection, Nice Proceedings 1976. Lecture Notes in Physics 71, Springer-Verlag, Berlin-Heidelberg-New York.
Chossat, P. [1979]. Bifurcation and stability of convective flows in a rotating or not rotating spherical shell. SIAM J. Appl. Math. 37, 624–647.
Chossat, P. [1982]. Le Problème de Bénard dans une Couche Sphèrique. Thèse d'État, Université de Nice.
Chossat, P. [1983]. Solutions avec symétrie diédrale dans les problèmes de bifurcation invariants par symétrie sphérique. C. R. Acad. Sci. Paris, Série I, 297, 639–642.
Chossat, P. & Lauterbach, R. [1989]. The instability of axisymmetric solutions in problems with spherical symmetry. SIAM J. Math. Anal. 20, 31–38.
Chow, S. N. & Lauterbach, R. [1988]. A bifurcation theorem for critical points of variational problems. Nonlinear Analysis, TMA 12, 51–61.
Cicogna, G. [1981]. Symmetry breakdown from bifurcation. Letters al Nuovo Cimento 31, 600–602.
Dancer, E. N. [1980]. On the existence of bifurcating solutions in the presence of symmetries. Proc. Roy. Soc. Edinburgh 85 A, 321–336.
Fiedler, B. & Mischaikow, K. [1990]. Dynamics of bifurcations for variational problems with O(3) equivariance: a Conley index approach, preprint.
Field, M. [1980]. Equivariant dynamical systems. Trans. A.M.S. 259, No. 1, 185–205.
Field, M. [1989]. Equivariant bifurcation theory and symmetry breaking. Dyn. Diff. Eqns., 1, 369–421.
Field, M. & Richardson, R. W. [1989]. Symmetry breaking and the maximal isotropy subgroup conjecture for finite reflection groups. Arch. Rational Mech. Anal. 105, No. 1, 61–94.
Field, M. & Richardson, R. W. [1990] Symmetry breaking in equivariant bifurcation problems. Bull. Amer. Math. Soc. 22, 79–84.
Friedricks, R. & Haken, H. [1986]. Static, wavelike and chaotic thermal convection in spherical geometries. Physical Review A 34, 2100–2120.
Golubitsky, M. & Guckenheimer, J. [1986]. Multiparameter Bifurcation Theory. Contemporary Math. 56, A.M.S., Providence.
Golubitsky, M. & Schaeffer, D. G. [1982]. Bifurcation with O(3) symmetry including applications to the Bénard problem. Commun. Pure Appl. Math. 35, 81–111.
Golubitsky, M. & Stewart, I. N. [1985]. Hopf bifurcation in the presence of symmetry. Arch. Rational Mech. Anal. 87, 107–165.
Golubitsky, M., Stewart, I. N. & Schaeffer, D. G. [1988]. Singularities and Groups in Bifurcation Theory, Vol. 2. Appl. Math. Sci. 69, Springer-Verlag, New York.
Ihrig, E. & Golubitsky, M. [1984]. Pattern selection with O(3) symmetry. Physica 13 D, 1–33.
Iooss, G. & Rossi, M. [1989]. Hopf bifurcation in the presence of spherical symmetry: analytical results. SIAM J. Math. Anal. 20, 511–532.
Knightly, G. H. & Sather, D. [1980]. Buckled states of a spherical shell under uniform external pressure. Arch. Rational Mech. Anal. 72, 315–380.
Krupa, M. [1988]. Bifurcations of critical group orbits. Ph. D. thesis, University of Houston.
Krupa, M. [1990]. Bifurcations of relative equilibria. SIAM J. Math. Anal., to appear.
Lang, S. [1978]. Algebra, Addison-Wesley.
Lauterbach, R. [1986]. An example of symmetry breaking with submaximal isotropy. In: Golubitsky & Guckenheimer [1986], 217–222.
Lauterbach, R. [1988]. Bifurcation with O(3)-symmetry. Habilitation Thesis, University of Augsburg.
Michel, M. [1972]. Nonlinear group action: Smooth action of compact Lie groups on manifolds. In: Statistical Mechanics and Field Theory, R. N. Sen & C. Weil, eds., Israel Univ. Press, Jerusalem, 133–150.
Michel, M. [1980]. Symmetry defects and broken symmetry configurations. Hidden symmetry, Rev. Mod. Phys. 52, 617–651.
Michel, M. [1989]. Private communication.
Miller, W. [1972]. Symmetry groups and their applications. Acad. Press, New York-London.
Milnor, J. [1968]. Singular points of complex hypersurfaces. Annals of Math. Studies, 61, Princeton Univ. Press, Princeton.
Le Mouel, J. M. [1975]. Traité de géophysique interre, Ch. 30. Massen, Paris.
Moutrane, E. [1988]. Intéraction de modes sphériques dans le problème de Bénard entre deux sphères. Thesis, Université de Nice.
Le Pichon, X. & Huchon, P. [1984]. Geoid, Pangea and convection. Earth Planet. Sci. Lett. 67, 123–135.
Riahi, N. [1984]. Nonlinear convection in a spherical shell. J. Phys. Soc. of Japan, 54, 2506–2512.
Roberts, P. H. [1965]. Convection in a self-gravitating fluid sphere. J. Math. 12, 128–137.
Sattinger, D. H. [1978]. Bifurcation from rotationally invariant states. J. Math. Phys. 19, No. 8, 1720–1732.
Sattinger, D. H. [1979]. Group Theoretic Methods in Bifurcation Theory. Lecture Notes in Math. 762. Springer-Verlag, Berlin.
Spiegel, E. A. & Zahn, J. P. [1977], eds. Problems of Stellar Convection. Lecture Notes in Phys. 71. Springer-Verlag, Berlin.
Vanderbauwhede, A. [1980]. Local Bifurcation and Symmetry. Habilitation Thesis, Rijksuniversiteit Gent.
Wigner, E. P. [1959]. Group Theory. Academic Press, New York.
Wilkinson, J. H. [1965]. The Algebraic Eigenvalue Problem. Oxford Univ. Press, Oxford.
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Chossat, P., Lauterbach, R. & Melbourne, I. Steady-State bifurcation with 0(3)-Symmetry. Arch. Rational Mech. Anal. 113, 313–376 (1991). https://doi.org/10.1007/BF00374697
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DOI: https://doi.org/10.1007/BF00374697