On the Hopf Bifurcation with Broken O(2) Symmetry

  • G. Dangelmayr
  • E. Knobloch
Part of the Springer Series in Synergetics book series (SSSYN, volume 37)


Translation and reflection symmetries introduce the group 0(2) into bifurcation problems with periodic boundary conditions. The effect on the Hopf bifurcation with 0(2)-symmetry of small terms breaking the translation symmetry is investigated. Two primary branches of standing waves are found. Secondary and tertiary bifurcations involving two different types of modulated waves are analyzed in the neighborhood of secondary Takens-Bogdanov bifurcations. The effects of breaking the phaseshift (in time) and reflection symmetries are briefly considered.


Manifold Shoe 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Ahlers and I. Rehberg, Phys. Rev. Lett. 56, 1373–1376 (1986)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    H.R. Brand, P.S. Lomdahl and A.C. Newell, Phys. Lett. A 118, 67–73 (1986)ADSGoogle Scholar
  3. 3.
    P. Chossat, C.R. de l’Académie des Sciences, Paris (1985)Google Scholar
  4. G. Dangelmayr and E. Knobloch, in preparationGoogle Scholar
  5. S. van Gils, preprintGoogle Scholar
  6. 6.
    M. Golubitsky and I. Stewart, Arch. Rat. Mech. Anal. 27, 107–165 (1985)MathSciNetCrossRefGoogle Scholar
  7. 7.
    J. Guckenheimer and P. Holmes, “Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields”, Springer 1983MATHGoogle Scholar
  8. R. Heinrichs, G. Ahlers and D.S. Cannell, preprintGoogle Scholar
  9. 9.
    E. Knobloch, Phys. Rev. A 34, 1538–1549 (1986)ADSCrossRefGoogle Scholar
  10. 10.
    E. Knobloch and M.R.E. Proctor, J. Fluid Mech. 108, 291–316 (1981)MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    E. Knobloch, in “Multiparameter Bifurcation Theory”, M. Golubitsky and J. Guckenheimer (eds.), Contemporary Mathematics 56, Amer. Math. Soc., Providence R.I. 1986Google Scholar
  12. E. Knobloch, A. Deane and J. Toomre, these proceedings.Google Scholar
  13. 13.
    P. Kolodner, A. Passner, C.M. Surko and R.W. Waiden, Phys. Rev. Lett. 56, 2621–2624 (1986)ADSCrossRefGoogle Scholar
  14. 14.
    E. Moses and V. Steinberg, Phys. Rev. A 34, 693–696 (1986)ADSCrossRefGoogle Scholar
  15. 15.
    W. Nagata, in “Multiparameter Bifurcation Theory”, M. Golubitsky and J. Guckenheimer (eds.), Contemporary Mathematics 56, Amer. Math. Soc, Providence R.I. 1986Google Scholar
  16. 16.
    D. Rand, Arch. Rat. Mech. Anal. 79, 1–38 (1982)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    I. Rehberg and G. Ahlers, Phys. Rev. Lett. 55, 500–504 (1985)ADSCrossRefGoogle Scholar
  18. 18.
    J.A. Sanders, Celestial Mechanics 28, 171–181 (1982)MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    R.W. Waiden, P. Kolodner, A. Passner and C.M. Surko, Phys. Rev. Lett. 55, 496–499 (1985)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • G. Dangelmayr
    • 1
  • E. Knobloch
    • 2
  1. 1.Institute for Information SciencesUniversity of TübingenTübingenFed. Rep. of Germany
  2. 2.Department of PhysicsUniversity of CaliforniaBerkeleyFed. Rep. of Germany

Personalised recommendations