Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-13T02:29:52.427Z Has data issue: false hasContentIssue false
  • English
  • Français

Degenerate bifurcations near a double eigenvalue in the Brusselator

Published online by Cambridge University Press:  17 February 2009

Gerhard Dangelmayr
Gerhard Dangelmayr
Affiliation:
Institute for Information Science, University of Tübingen, Tübingen, Federal Republic of Germany.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The steady state bifurcations near a double zero eigenvalue of the reaction diffusion equation associated with a tri-molecular chemical reaction (the Brusselator) are analysed. Special emphasis is put on three degeneracies where previous results of Schaeffer and Golubitsky do not apply. For these degeneracies it is shown by means of a LiapunovSchmidt reduction that the steady state bifurcations are determined by codimension-three normal forms. They are of types (9)31, (8)221 and (6a)ρ,κ in a recent classification of Z(2)-equivariant imperfect bifurcations with corank two. Each normal form couples an ordinary corank-1 bifurcation in the sense of Golubitsky and Schaeffer to a degenerate Z(2)-equivariant corank-1 bifurcation of Golubitsky and Langford in a specific way.

Type
Research Article
Information
The ANZIAM Journal , Volume 28 , Issue 4 , April 1987 , pp. 486 - 535
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Armbruster, D., Dangelmayr, G. and Güttinger, W., “Imperfection sensitivity of interacting Hopf and steady state bifurcations”, Phys. D 16 (1985), 99–123.CrossRefGoogle Scholar
[2]Auchmuty, J. F. G. and Nicolis, G., “Bifurcation analysis of non-linear reaction diffusion equations I”, Bull. Math. Biol. 37 (1975), 323–365.Google Scholar
[3]Chow, S. N. and Hale, J. K., Methods of bifurcation theory (Springer, Berlin, 1982 1982).CrossRefGoogle Scholar
[4]Dangelmayr, G. and Armbruster, D., “Classification of Z(2)-equivarient imperfect bifurcations with corank two”, Proc. London Math. Soc., 46 (1983), 517–546.CrossRefGoogle Scholar
[5]Golubitsky, M. and Schaeffer, D., Singularities and groups in bifurcation theory, Vol. 1 (Springer, New York, 1985).CrossRefGoogle Scholar
[6]Golubitsky, M. and Schaffer, D., “Imperfect bifurcations in the presense of symmetry”, Comm. Math. Phys. 67 (1979), 205–232.CrossRefGoogle Scholar
[7]Golubitsky, M. and Langford, W. F., “Classification and unfoldings of degenerate Hopf bifurcations” J. Differential Equations 41 (1981), 375–415.CrossRefGoogle Scholar
[8]Herschkowitz-Kaufman, M., “Bifurcation analysis of non-linear reaction-diffusion equations II”, Bull. Math. Biol. 37 (1975), 589–636.CrossRefGoogle Scholar
[9]Keener, J. P., “Secondary bifurcation in non-linear reaction equations”, Stud. Appl. Math. 55 (1976), 187–211.CrossRefGoogle Scholar
[10]Lefever, R. and Prigogine, I., “Symmetry breaking instabilities in dissipative systems”, J. Chem. Phys. 48 (1968), 1695–1700.Google Scholar
[11]Schaeffer, D. and Golubitsky, M., “Bifurcation analysis near a double degenerate eigenvalue of a model chemical reaction”, Arch. Rational Mech. Anal. 75 (1981), 315–347.CrossRefGoogle Scholar
[12]Vainberg, M. M. and Trenogin, V. A., Theory of branching of solutions of nonlinear equations (Noordhoff, Leyden, 1974).Google Scholar