Structurally Stable Bifurcations in Optical Bistability

  • D. Armbruster
  • G. Dangelmayr
Part of the NATO ASI Series book series (NSSB, volume 116)


The equations encountered in optical bistability are solvable analytically only for limiting cases (mean field limit, pure absorptive case etc.). Here the question arises: are these solutions and their types of bifurcations persistent when moving away from the limits? An answer to this question of structural stability is given by imperfect bifurcation theory (developed by Golubitsky & Schaeffer)1. The aim of imperfect bifurcation theory is the classification of degenerate bifurcation problems. Bifurcation means a qualitative change of the behaviour of the solutions of nonlinear equations (p.d.e., o.d.e., algebraic) when a distinguished physical parameter is varied. This corresponds to the fact that experimental graphs are usually plotted against a single parameter (in our case this is the incident field EI). Classification is done by means of an equivalence relation, which puts all bifurcation problems that behave qualitatively similar (i.e., exhibit the same stability changes) into the same class described by a polynomial normal form. Two classes differ by the number and type of perturbations which change the bifurcation qualitatively. These perturbations are called unfoldings and their number is the codimension of the problem.


Hopf Bifurcation Bifurcation Diagram Bifurcation Point Order Transition Optical Bistability 
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.
    M. Golubitsky and D. Schaeffer, Comm. Pure & Appl. Math. 32:21 (1979).MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    G. Dangelmayr and D. Armbruster, Proc. London Math. Soc. (3), 46:517 (1983).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    D. Armbruster, G. Dangelmayr and W. Güttinger, Imperfection sensitivity of interacting Hopf and steady-state bifurcations, preprint Tübingen (1982), submitted for publication.Google Scholar
  4. 4.
    D. Armbruster, An organizing centre for optical bistability and self-pulsing, submitted for publication.Google Scholar
  5. 5.
    D. Armbruster and G. Dangelmayr, An organizing centre for polarization switching, to be published.Google Scholar
  6. 6.
    M. Golubitsky and W.F. Langford, J. Diff. Equs. 41:375 (1981).MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    G.P. Agrawal and H.J. Carmichael, Phys.Rev.A. 19:2074 (1979).MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    L.A. Lugiato, Optics Comm. 33:108 (1980).ADSCrossRefGoogle Scholar
  9. 9.
    M.W. Hamilton, R.J. Ballagh and W.J. Sandle, Z. Phys. B 49:263 (1982).ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1984

Authors and Affiliations

  • D. Armbruster
    • 1
  • G. Dangelmayr
    • 1
  1. 1.Institute for Information SciencesUniversity of TübingenTübingenGermany

Personalised recommendations