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Hopf Bifurcation in Anisotropic Systems

  • Gerhard Dangelmayr
  • Michael Wegelin
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 115)

Abstract

Oscillation patterns predicted by the Hopf bifurcation with the symmetries O(2) × O(2), D m × O(2) and D m × D n are reviewed and discussed in the context of spatially continuous and discrete systems.

Keywords

Periodic Solution Normal Form Hopf Bifurcation Center Manifold Oscillation Pattern 
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

  1. [I]
    F. AMDJADI, P.J. ASTON AND P. PLECHAC, Symmetry breaking Hopf bifurcations in equations with O(2) symmetry with application to the Kuramoto-Sivashinsky equation, J. Comp. Physics, 131, 181–192, (1997).MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    G. DANGELMAYR, W. GüTTINGER AND M. WEGELIN, Hopf bifurcation with D 3 × D 3 -symmetry, Zeitschrift für angewandte Mathematik und Physik (ZAMP), 44, 595–638, (1993).zbMATHCrossRefGoogle Scholar
  3. [3]
    G. DANGELMAYR, W. GüTTINGER, J. OPPENLäNDER, J. TOMES AND M. WEGELIN, Synchronized patterns in hierarchical networks of neural oscillators, Physica D, 121, 213–232, (1998).MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    G. DANGELMAYR, J. HETTEL AND E. KNOBLOCH, Parity-breaking bifurcation in inhomogenous systems, Nonlinearity, 10, 1093–1114, (1997).MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    G. DANGELMAYR AND M. WEGELIN, in preparation.Google Scholar
  6. [6]
    B. DIONNE, M. GOLUBITSKY AND I. STEWART, Coupled cells with internal symmetry, Part I: wreath products, Nonlinearity, 9, 559–574, (1996).MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    B. DIONNE, M. GOLUBITSKY AND I. STEWART, Coupled cells with internal symmetry, Part II: direct products, Nonlinearity, 9, 575–600, (1996).MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    M. GOLUBITSKY, V.G. LEBLANC AND I. MELBOURNE, Meandering of the spiral tip: an alternative approach, J. Nonlin. Sci., 7, 557–586, (1997).MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    M. GOLUBITSKY AND I. STEWART, Hopf bifurcation in the presence of symmetry, Arch. Rat. Mech. Anal., 87, 107–165, (1985).MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    M. GOLUBITSKY AND I. STEWART, Hopf bifurcation with dihedral group symmetry, in M. Golubitsky and J. Guckenheimer (eds.): Multiparameter Bifurcation Theory, Contemporary Mathematics Vol. 56, Providence, AMS, 131–173, (1986).Google Scholar
  11. [11]
    M. GOLUBITSKY, I. STEWART AND D. SCHAEFFER, Singularities and Groups in Bifurcation Theory, Vol. bf II, Springer, (1986).Google Scholar
  12. [12]
    E. KNOBLOCH AND J. DELUCA, Amplitude equations for travelling wave convection, Nonlinearity, 2, 975–980, (1990).MathSciNetCrossRefGoogle Scholar
  13. [13]
    M. KRUPA, Bifurcations of relative equilibria, SIAM J. Math. Anal., 21, 1453–1486, (1990).MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    A. MIELKE, A spatial center manifold approach to steady state bifurcations from spatially periodic patterns, Chapter 4, in G. Dangelmayr, B. Fiedler, K. Kirchgässner and A. Mielke: Dynamics of Nonlinear Waves in Dissipative Systems, Pitman Research Notes in Mathematics, Vol. 352, Longman, (1996).Google Scholar
  15. [15]
    M. SILBER, H. RIECKE AND L. KRAMER, Symmetry breaking Hopf bifurcation in anisotropic systems, Physica D, 61, 260–278, (1992).MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    M. WEGELIN, SC.D. thesis, University of Tübingen, (1993).Google Scholar
  17. [17]
    M. WEGELIN, Hopf bifurcation in symmetrically coupled lasers, in P. Chossat (ed.): Dynamics, Bifurcation and Symmetry, Kluwer, 343–354, (1994).Google Scholar
  18. [18]
    D. WOODS, Three coupled oscillators with internal Z 2 symmetries, Nonlinearity, to appear.Google Scholar
  19. [19]
    D. WOODS, Coupled oscillators with internal symmetries, PhD. thesis, Mathematics Institute, University of Warwick, (1995).Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Gerhard Dangelmayr
    • 1
  • Michael Wegelin
    • 1
  1. 1.Department of MathematicsColorado State UniversityFort CollinsUSA

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