Abstract
Translation and reflection symmetries introduce the group 0(2) into bifurcation problems with periodic boundary conditions. The effect on the Hopf bifurcation with 0(2)-symmetry of small terms breaking the translation symmetry is investigated. Two primary branches of standing waves are found. Secondary and tertiary bifurcations involving two different types of modulated waves are analyzed in the neighborhood of secondary Takens-Bogdanov bifurcations. The effects of breaking the phaseshift (in time) and reflection symmetries are briefly considered.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. Ahlers and I. Rehberg, Phys. Rev. Lett. 56, 1373–1376 (1986)
H.R. Brand, P.S. Lomdahl and A.C. Newell, Phys. Lett. A 118, 67–73 (1986)
P. Chossat, C.R. de l’Académie des Sciences, Paris (1985)
G. Dangelmayr and E. Knobloch, in preparation
S. van Gils, preprint
M. Golubitsky and I. Stewart, Arch. Rat. Mech. Anal. 27, 107–165 (1985)
J. Guckenheimer and P. Holmes, “Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields”, Springer 1983
R. Heinrichs, G. Ahlers and D.S. Cannell, preprint
E. Knobloch, Phys. Rev. A 34, 1538–1549 (1986)
E. Knobloch and M.R.E. Proctor, J. Fluid Mech. 108, 291–316 (1981)
E. Knobloch, in “Multiparameter Bifurcation Theory”, M. Golubitsky and J. Guckenheimer (eds.), Contemporary Mathematics 56, Amer. Math. Soc., Providence R.I. 1986
E. Knobloch, A. Deane and J. Toomre, these proceedings.
P. Kolodner, A. Passner, C.M. Surko and R.W. Waiden, Phys. Rev. Lett. 56, 2621–2624 (1986)
E. Moses and V. Steinberg, Phys. Rev. A 34, 693–696 (1986)
W. Nagata, in “Multiparameter Bifurcation Theory”, M. Golubitsky and J. Guckenheimer (eds.), Contemporary Mathematics 56, Amer. Math. Soc, Providence R.I. 1986
D. Rand, Arch. Rat. Mech. Anal. 79, 1–38 (1982)
I. Rehberg and G. Ahlers, Phys. Rev. Lett. 55, 500–504 (1985)
J.A. Sanders, Celestial Mechanics 28, 171–181 (1982)
R.W. Waiden, P. Kolodner, A. Passner and C.M. Surko, Phys. Rev. Lett. 55, 496–499 (1985)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dangelmayr, G., Knobloch, E. (1987). On the Hopf Bifurcation with Broken O(2) Symmetry. In: Güttinger, W., Dangelmayr, G. (eds) The Physics of Structure Formation. Springer Series in Synergetics, vol 37. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-73001-6_30
Download citation
DOI: https://doi.org/10.1007/978-3-642-73001-6_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-73003-0
Online ISBN: 978-3-642-73001-6
eBook Packages: Springer Book Archive