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Dynamics near steady state bifurcations in problems with spherical symmetry

  • Reiner Lauterbach
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1463)

Abstract

We give a complete description of the dynamics near a bifurcation point where spontaneous symmetry breaking from an O(3) invariant state occurs. The main hypotheses is that the kernel of the linearized equation is the (natural) irreducible seven dimensional representation of O(3).

Keywords

Irreducible Representation Unstable Manifold Heteroclinic Orbit Isotropy Subgroup Maximal Isotropy 
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References

  1. Chossat, P. [1979]: Bifurcation and stability of convective flows in a rotating or nonrotating spherical shell, SIAM J. Appl. Math. 37, 624–647MathSciNetCrossRefzbMATHGoogle Scholar
  2. Chossat, P. & Lauterbach, R. [1989]: The instability of axisymmetric solutions in problems with spherical symmetry, SIAM J. Appl. Anal. 20(1), 31–38MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chossat, P. & Lauterbach, R. & Melbourne, I. [1990]: Steady sate bifurcation with O(3) symmetry, Arch. Rat. Mech. and Anal. (in press)Google Scholar
  4. Chow, S.-N. & Lauterbach, R. [1988]: A bifurcation theorem for critical points of variational problems, Nonl. Anal., TMA 12, 51–61MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cicogna, G. [1981]: Symmetry breakdown from bifurcation. Lettere el Nuovo Cimente, 31, 600–602MathSciNetCrossRefGoogle Scholar
  6. Fiedler, B. & Mischaikov, K. [1989]: Dynamics of bifurcations for variational problems with O(3) equivariance: a Conley index approach, Preprint SFB 123, No. 536Google Scholar
  7. Golubitsky, M., Stewart, I. & Schaeffer, D.G. [1988]: Singularities and groups in bifurcation theory, Vol. II, Springer Verlag, Heidelberg New YorkCrossRefzbMATHGoogle Scholar
  8. Henry, D. [1981]: Geometric theory of semilinear parabolic equations, Springer Lecture Notes 840, Springer Verlag, New York-HeidelbergzbMATHGoogle Scholar
  9. Ihrig, E. & Golubitsky, M. [1984]: Pattern selection with O(3) symmetry, Physica 13D, 1–33MathSciNetzbMATHGoogle Scholar
  10. Kato, T. [1976]: Perturbation theory for linear operators, Springer Verlag, New York-HeidelbergCrossRefzbMATHGoogle Scholar
  11. Knightly, G.H. and Sather, D. [1980]: Buckled states of a spherical shell under uniform external pressure, Arch. Rat. Mech. and Anal. 72, 315–380MathSciNetCrossRefzbMATHGoogle Scholar
  12. Lauterbach, R. [1988]: Problems with spherical symmetries: studies on bifurcations and dynamics for O(3)-equivariant equations. Habilitationsschrift, Univ. AugsburgGoogle Scholar
  13. Lauterbach, R. [1989]: Maximal isotropy subgroups and bifurcation: an example, PreprintGoogle Scholar
  14. Sattinger, D.H. [1979]: Group theoretic methods in bifurcation theory. Springer Lecture Notes 762, Springer Verlag, New York-HeidelbergzbMATHGoogle Scholar
  15. Vanderbauwhede, A. [1982]: Local bifurcation and symmetry, Research Notes in Mathematics 75, Pitman, BostonzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Reiner Lauterbach

There are no affiliations available

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