Eigenvalue Problems with the Symmetry of a Group and Bifurcations

  • B. Werner
Part of the NATO ASI Series book series (ASIC, volume 313)


For parameter dependent nonlinear Γ-equivariant dynamical systems
$$\dot x = g(x,\lambda ),\,g:X \times IR \to X,\,X = I{R^n},$$
, along a branch of Γ-symmetric equilibria {(x(s),λ(s)) : sI} the Jacobians \( A(s): = \tfrac{{\partial g}}{{\partial x}}(x(s),\lambda (s)) \)share the Γ-equivariance with g(.,λ):
$$\gamma A(s) = A(s)\gamma \,for\,all\,\gamma \, \in \,\Gamma $$
.(Here Γ is a compact Lie group of orthogonal nx n-matrices).


Irreducible Representation Hopf Bifurcation Bifurcation Point Complex Type Simple Eigenvalue 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Bossavit. Symmetry, groups and boundary value problems. A progressive introduction to noncommutative harmonic analysis of partial differential equations in domains with geometrical symmetry. Computer Meth. in Appl. Mech. and Eng. 56, 167–215, 1986.Google Scholar
  2. [2]
    G. Dangelmayr, E. Knobloch. The Takens-Bogdanov Bifurcation with O(2)-Symmetry. Phil. Trans. R. Soc. Lond. A 322, 243–279, 1987.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    M. Dellnitz. Hopf-Verzweigung in Systemen mit Symmetrie und deren Numerische Behandlung. Wissenschaftliche Beiträge aus Europäischen Hochschulen. Verlag an der Lottbek. ( Dissertation Universität Hamburg ) 1989.Google Scholar
  4. [4]
    M. Dellnitz, B. Werner. Computational methods for bifurcation problems with symmetries — with special attention to steady state and Hopf bifurcation points. J. of Comp, and Appl. Math. 26, 97–123, 1989. (special issue on Continuation Techniques and Bifurcation Problems(H.D. Mittelmann, D. Roose (eds))MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    B. Fiedler. Global Hopf bifurcation of two-parameter flows. Arch. Rat. Mech. and Anal. 94, 59–81, 1986.MathSciNetCrossRefGoogle Scholar
  6. [6]
    M. Golubitsky, I. Stewart, D. Schaeffer. Singularities and Groups in Bifurcation Theory, Vol. 2, Springer 1988.Google Scholar
  7. [7]
    M. Golubitsky, I. Stewart. Hopf bifurcation with dihedral group symmetry: coupled nonlinear oscillators. In Multiparameter Bifurcation Theory, M. Golubitsky, J. Guckenheimer (eds.). Contemporary Mathematics 56, 131–173. Amer. Math. Soc., Providence, 1986.Google Scholar
  8. [8]
    A. Griewank, G. Reddien. Characterization and computation of generalized turning points. SIAM J. Numer. Anal. 21, 176–185, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    J. Guckenheimer, P. Holmes. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New-York, 1983.zbMATHGoogle Scholar
  10. [10]
    T. Healey. A group-theoretic approach to computational bifurcation with symmetry. Computer Methods in Appl. Mech. and Eng. 67, 257–295, 1988.Google Scholar
  11. [11]
    J.P. Serre Linear representations of finite groups. Springer 1977.Google Scholar
  12. [12]
    I. Schreiber, M. Holodniok, M. Kubicek, M. M.rek. Periodic and aperiodic regimes in coupled dissipative chemical oscillators. J. of Statist. Phys. 43, 489–518, 1986.MathSciNetGoogle Scholar
  13. [13]
    E. Stiefel, A. Fässler Gruppentheoretische Methoden und ihre Anwendung. Teubner 1979.Google Scholar
  14. [14]
    P. Stork. Statische Verzweigungen mit Symmetrien. Diplomarbeit. Institut für Angewandte Mathematik der Universität Hamburg 1989.Google Scholar
  15. [15]
    A. Vanderbauwhede. Local Bifurcation Theory and Symmetry. Pitman 1982.Google Scholar
  16. [16]
    B. Werner. Computational methods for bifurcation problems with symmetries and applications to steady states of n-box reaction-diffusion models. In Numerical Analysis 1987., D. F. Griffiths, G. A. Watson (eds.), 279–293, Pitman 1988.Google Scholar
  17. [17]
    B. Werner, A. Spence. The computation of symmetry-breaking bifurcation points. SIAM J. Numer. Anal. 21, 388–399, 1984.zbMATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • B. Werner
    • 1
  1. 1.Institut für Angewandte MathematikUniversität HamburgHamburg 13Federal Republic of Germany

Personalised recommendations