Boundary conditions as symmetry constraints

  • J. D. Crawford
  • M. Golubitsky
  • M. G. M. Gomes
  • E. Knobloch
  • I. N. Stewart
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1463)


Fujii, Mimura, and Nishiura [1985] and Armbruster and Dangelmayr [1986, 1987] have observed that reaction-diffusion equations on the interval with Neumann boundary conditions can be viewed as restrictions of similar problems with periodic boundary conditions; and that this extension reveals the presence of additonal symmetry constraints which affect the generic bifurcation phenomena. We show that, more generally, similar observations hold for multi-dimensional rectangular domains with either Neumann or Dirichlet boundary conditions, and analyse the group-theoretic restrictions that this structure imposes upon bifurcations. We discuss a number of examples of these phenomena that arise in applications, including the Taylor-Couette experiment, Rayleigh-Bénard convection, and the Faraday experiment.


Neumann Boundary Condition Mode Number Isotropy Subgroup Pitchfork Bifurcation Bifurcation Problem 
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. C.D. Andereck, S.S. Liu, and H.L. Swinney [1986]. Flow regimes in a circular Couette system with independently rotating cylinders, J. Fluid Mech. 164, 155–183.CrossRefGoogle Scholar
  2. D. Armbruster and G. Dangelmayr [1987]. Coupled stationary bifurcations in nonflux boundary value problems, Math. Proc. Camb. Phil. Soc. 101 167–192.MathSciNetCrossRefzbMATHGoogle Scholar
  3. T.B. Benjamin [1978]. Bifurcation phenomena in steady flows of a viscous fluid, Proc. R. Soc. London A 359 1–26, 27–43.MathSciNetCrossRefGoogle Scholar
  4. T.B. Benjamin and F. Ursell [1954]. The stability of the plane free surface of a liquid in vertical periodic motion, Proc. R. Soc. London A 255 505–517.MathSciNetCrossRefzbMATHGoogle Scholar
  5. P. Chossat, Y. Demay, and G. Iooss [1987]. Interactions des modes azimutaux dans le problème de Couette-Taylor, Arch. Rational Mech. Anal. 99 213–248.MathSciNetCrossRefGoogle Scholar
  6. S. Ciliberto and J. Gollub [1985a]. Chaotic mode competition in parametrically forced surface waves, J. Fluid Mech. 158 381–398.MathSciNetCrossRefGoogle Scholar
  7. S. Ciliberto and J. Gollub [1985b]. Phenomenological model of chaotic mode competition in surface waves, Nuovo Cimento 60 309–316.CrossRefGoogle Scholar
  8. J.D.Crawford, M.Golubitsky, and E.Knobloch [1989]. In preparation.Google Scholar
  9. J.D. Crawford, E. Knobloch, and H. Riecke [1989]. Competing parametric instabilities with circular symmetry, Phys. Lett A 135 20–24.MathSciNetCrossRefGoogle Scholar
  10. G. Dangelmayr and D. Armbruster [1986]. Steady state mode interactions in the presence of O(2) symmetry and in non-flux boundary conditions. In Multiparameter Bifurcation Theory (eds. M. Golubitsky and J. Guckenheimer), Contemp. Math. 56, Amer. Math. Soc., Providence.CrossRefGoogle Scholar
  11. G. Dangelmayr and E. Knobloch [1987]. The Takens-Bogdanov bifurcation with O(2) symmetry, Phil. Trans. R. Soc. Lond. A322 243–279.MathSciNetCrossRefzbMATHGoogle Scholar
  12. P.G. Drazin [1975]. On the effects of sidewalls on Bénard convection, Z. angew. Math. Phys. 27, 239–243.CrossRefzbMATHGoogle Scholar
  13. Z.C. Feng and P.R. Sethna [1989]. Symmetry-breaking bifurcations in resonant surface waves, J. Fluid Mech. 199 495–518.MathSciNetCrossRefzbMATHGoogle Scholar
  14. M.J.Field, M.Golubitsky, and I.N.Stewart [1990]. Bifurcations on hemispheres, preprint, Univ. of Houston.Google Scholar
  15. H. Fujii, M. Mimura, and Y. Nishiura [1982]. A picture of the global bifurcation diagram in ecological interacting and diffusing systems, Physica 5D 1–42.MathSciNetGoogle Scholar
  16. M. Golubitsky and W.F. Langford [1988]. Pattern formation and bistability in flow between counterrotating cylinders, Physica 32D 362–392.MathSciNetzbMATHGoogle Scholar
  17. M. Golubitsky, J.E. Marsden, and D.G. Schaeffer [1984]. Bifurcation problems with hidden symmetries, in Partial Differential Equations and Dynamical Systems (ed. W.E. Fitzgibbon III), Research Notes in Math. 101, Pitman, San Francisco, 181–210.Google Scholar
  18. M. Golubitsky and I.N. Stewart [1986]. Symmetry and stability in Taylor-Couette flow, SIAM J. Math. Anal. 17 249–288.MathSciNetCrossRefzbMATHGoogle Scholar
  19. M. Golubitsky, I.N. Stewart, and D.G. Schaeffer [1988]. Singularities and Groups in Bifurcation Theory vol. II, Applied Math. Sci. 69, Springer, New York.CrossRefzbMATHGoogle Scholar
  20. M.G.M. Gomes [1989]. Steady-state mode interactions in rectangular domains, M.Sc. thesis, Univ. of Warwick.Google Scholar
  21. P. Hall and I.C. Walton [1977]. The smooth transition to a convective régime in a two-dimensional box, Proc. R. Soc. Lond. A 358 199–221.MathSciNetCrossRefzbMATHGoogle Scholar
  22. L.M. Hocking [1987]. The damping of capillary-gravity waves at a rigid boundary, J. Fluid Mech. 179 253–266.CrossRefzbMATHGoogle Scholar
  23. G.W. Hunt [1982]. Symmetries of elastic buckling, Eng. Struct. 4 21–28.CrossRefGoogle Scholar
  24. G. Iooss [1986]. Secondary bifurcations of Taylor vortices into wavy inflow and outflow boundaries, J. Fluid Mech. 173 273–288.CrossRefzbMATHGoogle Scholar
  25. E. Meron and I. Procaccia [1986]. Low dimensional chaos in surface waves: theoretical analysis of an experiment, Phys. Rev. A 34 3221–3237.CrossRefGoogle Scholar
  26. T. Mullin [1982]. Cellular mutations in Taylor flow, J. Fluid Mech. 121 207–218.CrossRefGoogle Scholar
  27. D.G. Schaeffer [1980]. Qualitative analysis of a model for boundary effects in the Taylor problem, Math. Proc. Camb. Phil. Soc. 87 307–337.MathSciNetCrossRefzbMATHGoogle Scholar
  28. M. Silber and E. Knobloch [1989]. Parametrically excited surface waves in square geometry, Phys. Lett. A 137 349–354.MathSciNetCrossRefGoogle Scholar
  29. F. Simonelli and J. Gollub [1989]. Surface wave mode interactions: effects of symmetry and degeneracy, J. Fluid Mech. 199 471–494.MathSciNetCrossRefGoogle Scholar
  30. G.I. Taylor [1923]. Stability of a viscous liquid contained between two rotating cylinders, Phil. Trans. R. Soc. Lond. A 223 289–343.CrossRefzbMATHGoogle Scholar
  31. M. Umeki and T. Kambe [1989]. Nonlinear dynamics and chaos in parametrically excited surface waves, J. Phys. Soc. Japan 58 140–154.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • J. D. Crawford
  • M. Golubitsky
  • M. G. M. Gomes
  • E. Knobloch
  • I. N. Stewart

There are no affiliations available

Personalised recommendations