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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)

Abstract

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.

Keywords

Hopf Bifurcation Standing Wave Bifurcation Diagram Center Manifold Primary Branch 
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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

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