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Dynamics near steady state bifurcations in problems with spherical symmetry

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

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  • Chossat, P. [1979]: Bifurcation and stability of convective flows in a rotating or nonrotating spherical shell, SIAM J. Appl. Math. 37, 624–647

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Chossat, P. & Lauterbach, R. [1989]: The instability of axisymmetric solutions in problems with spherical symmetry, SIAM J. Appl. Anal. 20(1), 31–38

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Chossat, P. & Lauterbach, R. & Melbourne, I. [1990]: Steady sate bifurcation with O(3) symmetry, Arch. Rat. Mech. and Anal. (in press)

    Google Scholar 

  • Chow, S.-N. & Lauterbach, R. [1988]: A bifurcation theorem for critical points of variational problems, Nonl. Anal., TMA 12, 51–61

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Cicogna, G. [1981]: Symmetry breakdown from bifurcation. Lettere el Nuovo Cimente, 31, 600–602

    CrossRef  MathSciNet  Google Scholar 

  • Fiedler, B. & Mischaikov, K. [1989]: Dynamics of bifurcations for variational problems with O(3) equivariance: a Conley index approach, Preprint SFB 123, No. 536

    Google Scholar 

  • Golubitsky, M., Stewart, I. & Schaeffer, D.G. [1988]: Singularities and groups in bifurcation theory, Vol. II, Springer Verlag, Heidelberg New York

    CrossRef  MATH  Google Scholar 

  • Henry, D. [1981]: Geometric theory of semilinear parabolic equations, Springer Lecture Notes 840, Springer Verlag, New York-Heidelberg

    MATH  Google Scholar 

  • Ihrig, E. & Golubitsky, M. [1984]: Pattern selection with O(3) symmetry, Physica 13D, 1–33

    MathSciNet  MATH  Google Scholar 

  • Kato, T. [1976]: Perturbation theory for linear operators, Springer Verlag, New York-Heidelberg

    CrossRef  MATH  Google Scholar 

  • Knightly, G.H. and Sather, D. [1980]: Buckled states of a spherical shell under uniform external pressure, Arch. Rat. Mech. and Anal. 72, 315–380

    CrossRef  MathSciNet  MATH  Google Scholar 

  • Lauterbach, R. [1988]: Problems with spherical symmetries: studies on bifurcations and dynamics for O(3)-equivariant equations. Habilitationsschrift, Univ. Augsburg

    Google Scholar 

  • Lauterbach, R. [1989]: Maximal isotropy subgroups and bifurcation: an example, Preprint

    Google Scholar 

  • Sattinger, D.H. [1979]: Group theoretic methods in bifurcation theory. Springer Lecture Notes 762, Springer Verlag, New York-Heidelberg

    MATH  Google Scholar 

  • Vanderbauwhede, A. [1982]: Local bifurcation and symmetry, Research Notes in Mathematics 75, Pitman, Boston

    MATH  Google Scholar 

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© 1991 Springer-Verlag

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Lauterbach, R. (1991). Dynamics near steady state bifurcations in problems with spherical symmetry. In: Roberts, M., Stewart, I. (eds) Singularity Theory and its Applications. Lecture Notes in Mathematics, vol 1463. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085434

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  • DOI: https://doi.org/10.1007/BFb0085434

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53736-6

  • Online ISBN: 978-3-540-47047-2

  • eBook Packages: Springer Book Archive