Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-25T12:27:32.078Z Has data issue: false hasContentIssue false
  • English
  • Français

Coupled stationary bifurcations in non-flux boundary value problems

Published online by Cambridge University Press:  24 October 2008

D. Armbruster  and
G. Dangelmayr
D. Armbruster
Affiliation:
Institute for Information Sciences, University of Tübingen, West Germany
G. Dangelmayr
Affiliation:
Institute for Information Sciences, University of Tübingen, West Germany
Get access Rights & Permissions [Opens in a new window]

Abstract

Coupled stationary bifurcations in nonlinear operator equations for functions, which are defined on a real interval with non-flux boundary conditions at the ends, are analysed in the framework of imperfect bifurcation theory. The bifurcation equations resulting from a Lyapunov–Schmidt reduction possess a natural structure which can be obtained by taking real parts of a diagonal action in ℂ2 of the symmetry group 0(2). A complete unfolding theory is developed and bifurcation equations are classified up to codimension two. Structurally stable bifurcation diagrams are given and their dependence on the wave numbers of the unstable modes is clarified.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 1 , January 1987 , pp. 167 - 192
Copyright
Copyright © Cambridge Philosophical Society 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]D., Armbruster. Constructing universal unfoldings using Gröbner bases. (Preprint, Tubingen, 1985.)Google Scholar
[2]D., Armbruster and G., Dangelmayr. Corank two bifurcations for the Brusselator with non-flux boundary conditions. (Preprint, Tubingen, 1986.)Google Scholar
[3]B., Buchberger. In Recent Trends in Multidimensional System Theory, ed. Bose, N. K. (Reidel, 1985).Google Scholar
[4]E., Buzano and A., Russo. Non-axisymmetric post-buckling behaviour of a complete thin cylindrical shell under axial compression. (Preprint, Trieste, 1984.)Google Scholar
[5]Dangelmayr, G.. Steady-state mode interactions in the presence of 0(2)-symmetry. To appear in Dynamics and Stability of Systems.Google Scholar
[6]G., Dangelmayr and D., Armbruster. Steady-state mode interactions in the presence of 0(2)-symmetry and in non-flux boundary value problems. To appear in Proc. AMS-Gonference on Multiparameter Bifurcation Theory (Arcata, 1985).Google Scholar
[7]G., Dangelmayr and D., Armbruster. Classification of Z(2)-equivariant imperfect bifurcations with corank 2. Proc. London Math. Soc. 46 (3), 517 (1983).Google Scholar
[8]Th., Erneux and Reiss, E. O.. Singular secondary bifurcations. SIAM J. Appl. Math. 44 (3), 463 (1984).Google Scholar
[9]H., Fuji, M., Mimura and Y., Nishiura. A picture of the global bifurcation diagram in ecological interacting and diffusing systems. Physica 5 D, 1 (1982).Google Scholar
[10]R., Gebauer and H., Kredel. The Buchberger algorithm system. Sigsam 18 (1) (1984).Google Scholar
[11]M., Golubitsky and D., Schaeffer. Singularities and Groups in Bifurcation Theory, vol. I (Springer, 1985).Google Scholar
[12]M., Golubitsky and D., Schaeffer. A theory for imperfect bifurcations via singularity theory. Comm. Pure Appl. Math. 32, 21 (1979).Google Scholar
[13]M., Golubitsky and Schaeffer, D.. Imperfect bifurcations in the presence of symmetry. Comm. Math. Phys. 67, 205 (1979).Google Scholar
[14]M., Golubitsky and Lanoford, W. F.. Classification and unfolding of degenerate Hopf bifurcations. J. Differential Equations 41 (3), 375 (1981).Google Scholar
[15]M., Golubitsky, J., Marsden and D., Schaeffer. Bifurcation problems with hidden symmetries. In Partial Differential Equations and Dynamical Systems, ed. Fitzgibbon, W. (Pitman, 1984).Google Scholar
[16]H., Kredel. On a generalization of the Gauss' algorithm to modules over polynomial rings. (Preprint, Heidelberg, 1985.)Google Scholar