Abstract
In these notes, our goal is to illustrate why blowing-up is a powerful technique in equivariant bifurcation theory. In addition, we shall explore some new directions that we believe may lead to more effective computational and analytical methods in the theory. Most of the work that we describe here has arisen out of a study of the effect of (high order) symmetry breaking perturbations on branches of relative equilibria [9].
This is a preview of subscription content, log in via an institution to check access.
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
D Armbruster, J Guckenheimer and P Holmes. ‘Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry’, Physica D,(1988), 257–282.
J Bochnak, M Coste and M-F Roy. Géométrie algébrique réelle (Ergebnisse der Mathematik and ihrer Grenzgebiete; Folge 3, Bd. 2, Springer-Verlag, Berlin, Heidelberg, New York, 1987.)
P Chossat. ‘Forced reflectional symmetry breaking of an O(2)-symmetric homoclinic cycle’, Nonlinearity, 6(5), (1993), 723–732.
P Chossat and M J Field. ‘Geometric analysis of the effect of symmetry breaking perturbations on an O(2) invariant homoclinic cycle’, In: Normal forms and Homoclinic chaos (W. F. Langford and W. Nagata, eds.) Fields Institute Communications, AMS, to appear.
M J Field. ‘Equivariant Dynamical Systems’, Trans. Amer. Math. Soc., 259(1980),185–205. 26(1982), 161–180.
M J Field. ‘Resolving actions of compact Lie groups’, Bull. Austral. Math. Soc. 18(1978), 243–254.
M J Field, ‘Equivariant Bifurcation Theory and Symmetry Breaking’, J. Dynamics and Diff. Eqns.,1(4) (1989), 369–421.
M J Field, ‘Local structure of equivariant dynamics’ Singularities,Bifurcations, and Dynamics,Proceedings of Symposium on Singularity Theory and its Applications, Warwick, 1989 (eds. R. M. Roberts and I. N. Stewart), Lecture Notes in Mathematics 1463, Springer-Verlag, Heidelberg (1991), 168–195.
M J Field, ‘Symmetry breaking for compact Lie groups’, preprint, University of Houston, 1994.
M J Field. ‘Geometric methods in bifurcation theory’, In: Pattern formation and symmetry breaking in PDEs. Fields Institute Communications, AMS, to appear.
M J Field. ‘Determinacy and branching patterns for the equivariant Hopf bifurcation’, to appear in Nonlinearity 7, (1994).
M J Field and R W Richardson. ‘Symmetry Breaking and the Maximal Isotropy Subgroup Conjecture for Reflection Groups’, Arch. for Rational Mech. and Anal., 105(1) (1989), 61–94.
M J Field and R W Richardson. ‘Symmetry breaking and branching patterns in equivariant bifurcation theory I’, Arch. Rational Mech. Anal., 118(1992), 297–348.
M J Field and R W Richardson. ‘Symmetry breaking and branching patterns in equivariant bifurcation theory II’, Arch. Rational Mech. Anal., 120(1992), 147–490.
M J Field and J W Swift. ‘Hopf fibration and the Hopf bifurcation’, to appear in Nonlinearity 7, (1994).
M Golubitsky, I N Stewart and D G Schaeffer, Singularities and Groups in Bifurcation Theory,Vol. II,Applied Mathematical Sciences 69, Springer-Verlag, New York, Berlin, Heidelberg, 1988.
M Krupa, ‘Bifurcations of relative equilibria’, Siam J. Math. Anal., 21(6)(1990), 1453–1486.
R Lauterbach and M Roberts. ‘Heteroclinic cycles in Dynamical systems with broken spherical symmetry’, J. Diff. Eq., 100(1) (1992), 22–48.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Field, M. (1994). Blowing-Up in Equivariant Bifurcation Theory. In: Chossat, P. (eds) Dynamics, Bifurcation and Symmetry. NATO ASI Series, vol 437. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0956-7_10
Download citation
DOI: https://doi.org/10.1007/978-94-011-0956-7_10
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-4413-4
Online ISBN: 978-94-011-0956-7
eBook Packages: Springer Book Archive