Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-09-21T10:52:50.183Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Stability of Equivariant Bifurcation Problems and Their Unfoldings

Published online by Cambridge University Press:  20 November 2018

Ali Lari-Lavassani  and
Yung-Chen Lu
Ali Lari-Lavassani
Affiliation:
Department of Systems Design University of Waterloo Waterloo, Ontario
Yung-Chen Lu
Affiliation:
Department of Mathematics The Ohio State University Columbus, Ohio
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.

In their book Singularities and Groups in Bifurcation Theory M. Golubitsky, I. Stewart and D. Schaeffer have introduced an equivariant version of Martinet's notion of V (for variety)-equivalence with parameter. In this paper we give a unified proof that, in this context, infinitesimal stability is equivalent to stability at the local level of germs and that stability in the unfolding category is equivalent to versality.

Keywords

58E0758C27
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 35 , Issue 2 , 01 June 1992 , pp. 237 - 246
Copyright
Copyright © Canadian Mathematical Society 1992 

References

[AGV, 1985] Arnold, V. I., Gusein-Zade, S. M., and Varchenko, A. N., Singularities of Differentiable Maps, 1 Birkhàuser, Basel/Stuttgart, (1985).Google Scholar
[B, 1977] Bierstone, E., Generic equivariant maps. Real and Complex Singularities, Sijthoff and Nordhoff, (1977), 127-161.Google Scholar
[B, 1980] Bierstone, E., The Structure of Orbit Spaces and the Singularities of Equivariant Mappings, I.M.P.A.,Rio de Janeiro, 1980.Google Scholar
[D, 1984] Damon, J., The Unfolding and determinacy theorems for subgroups of Si and %_, Memoirs, A.M.S. 306, Providence, 1984.Google Scholar
[G, 1988] Gervais, J. J., Stability of Unfoldings in the Context of Equivariant Contact-Equivalence, Pacific Journal of Math. (2) 132(1988), 283–291.Google Scholar
[GSS, 1988] Golubitsky, M., Stewart, I., and Schaeffer, D. G., Singularities and Groups in Bifurcation Theory, 2, Springer, Berlin/Heidelberg/New York, 1988.Google Scholar
[1,1980] Izumiya, S., Stability of G-Unfoldings, Hokkaido Math. J. 9(1980), 36–45.Google Scholar
[L, 1990] Lari-Lavassani, A., Multiparameter Bifurcation With Symmetry Via Singularity Theory, Ph.D. Thesis, The Ohio State University, 1990.Google Scholar
[LL, 1990] Lari-Lavassani, A. and Lu, Y.-C., The Stability Theorem for Subgroups of SI and 3£, preprint, 1991.Google Scholar
[LL, 1991] Lari-Lavassani, A. and Lu, Y.-C., Equivariant Multiparameter Bifurcation Via Singularity Theory, preprint, 1991.Google Scholar
[M, 1975] Martinet, J., Déploiements versels des applications différentiables et classification des applicationsstables. In: Singularités d'Applications Différentiables, Plan-sur-Bex, Lecture Notes in Math. 535, Springer-Verlag, Berlin/Heidelberg/New York, (1975), 1–44.Google Scholar
[P, 1976] Poénaru, V., Singularités C°° en Présence de Symétrie, Lecture Notes in Math. 510, Springer-Verlag, Berlin/Heidelberg/New York, (1976).Google Scholar