Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T01:19:47.139Z Has data issue: false hasContentIssue false
  • English
  • Français

A perturbed Hopf bifurcation with reflection symmetry

Published online by Cambridge University Press:  14 November 2011

Wayne Nagata
Wayne Nagata
Affiliation:
Department of Mathematics, The University of British Columbia, 121-1984 Mathematics Rd, Vancouver, B. C. Canada V6T 1Y4
Get access Rights & Permissions [Opens in a new window]

Synopsis

We study the effects of a small symmetry breaking perturbation on a system of differential equations at a coupled Hopf bifurcation with O(2) symmetry, where the perturbation breaks the continuous rotation symmetry, but retains a reflection (Z2) symmetry. It is shown that for a large range of parameter values, the invariant manifolds of the unperturbed bifurcation persist and that for some values of normal form coefficients there are secondary bifurcations of nonsymmetric periodic standing wave solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 117 , Issue 1-2 , 1991 , pp. 1 - 20
Copyright
Copyright © Royal Society of Edinburgh 1991

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

1Dangelmayr, G. and Knobloch, E.. On the Hopf bifurcation with broken O(2) symmetry. In The Physics of Structure Formation: Theory and Simulation, eds. Giittinger, W. and Dangelmayr, G. pp. 387–393 (Berlin: Springer, 1987).CrossRefGoogle Scholar
2Gils, S. A. van and Mallet-Paret, J.. Hopf bifurcation and symmetry: travelling and standing waves on the circle. Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), 279–307.CrossRefGoogle Scholar
3Golubitsky, M., Stewart, I. and Schaeffer, D. G.. Singularities and Groups in Bifurcation Theory, Vol. 2 (New York: Springer, 1988).CrossRefGoogle Scholar
4Guckenheimer, J. and Holmes, P.Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (New York: Springer 1983).Google Scholar
5Hale, J. K.. Ordinary Differential Equations (Malabar: Krieger 1980).Google Scholar
6Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds, Lecture Notes in Mathematics 583 Berlin: Springer 1977).CrossRefGoogle Scholar
7Holmes, P., Marsden, J. and Scheurle, J.. Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations. In Hamiltonian Dynamical Systems, eds. Meyer, K. R. and Saari, D. G., Contemporary Mathematics 81, pp. 213–244 (Providence, R. I.: American Mathematical Society, 1988).Google Scholar
8Nagata, W.. Convection in a layer with sidewalls: bifurcation with reflection symmetries. Z. angew. Math. Phys., in press (1990).CrossRefGoogle Scholar