Hostname: page-component-7479d7b7d-qlrfm Total loading time: 0 Render date: 2024-07-13T19:18:59.339Z Has data issue: false hasContentIssue false
  • English
  • Français

Universal unfoldings in Banach spaces: reduction and stability

Published online by Cambridge University Press:  24 October 2008

R. J. Magnus
R. J. Magnus
Affiliation:
University of Iceland, Reykjavik
Get access Rights & Permissions [Opens in a new window]

Extract

This article is concerned with the bifurcation of critical points of a smooth real-valued function on a Banach space. There are two somewhat different aspects considered. The first, referred to as reduction, is a refinement of the Liapunov-Schmidt procedure by incorporating techniques from the theory of determinate germs and their universal unfoldings. A definite reduction procedure is given, whereby a germ or unfolding given in a Banach space may be replaced by a germ or unfolding in a finite-dimensional space in such a way that the geometry of the bifurcation set and the overlying critical point set is preserved. The aim is to provide a practicable tool, not an exhaustive theoretical discussion. How practicable it is may be seen from another paper (Magnus and Poston(7)) in which it is applied to a problem in elasticity; indeed, the present paper in part was originally the ‘machinery’ section of that paper. The second aspect, stability, is of more theoretical interest. The well known structural stability of universal unfoldings (see (5)) is extended to infinite-dimensions. Only a local theory is given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 1 , July 1979 , pp. 41 - 55
Copyright
Copyright © Cambridge Philosophical Society 1979

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

REFERENCES

(1)Bröcker, Th.Differentiate germs and catastrophes (Cambridge, Cambridge University Press, 1975).CrossRefGoogle Scholar
(2)Magnus, R. J.On universal unfoldings of certain real functions on a Banach space. Battelle-Geneva math, report no. 100. Proc. Cambridge Philos. Soc. 81 (1977), 91–95.CrossRefGoogle Scholar
(3)Magnus, R. J.Determinacy in a class of germs on a reflexive Banach space. Battelle-Geneva math, report no. 103. Proc. Cambridge Philos. Soc. 84 (1978), 293–302.CrossRefGoogle Scholar
(4)Michor, P. The division theorem in Banach spaces. Personal communication.Google Scholar
(5)Wasserman, G.Stability of unfoldings (Berlin, Springer-Verlag, 1974).CrossRefGoogle Scholar
(6)Zeeman, E. C. and Trotman, D. J. A. The classification of elementary catastrophes of codimension ≤ 5. Symposium on catastrophe theory, Seattle 1975 (Berlin, Springer-Verlag, 1976).Google Scholar
(7)Magnus, R. J. and Poston, T. On the full unfolding of the von Kármán equations at a double eigenvalue. Battelle-Geneva math, report no. 109.Google Scholar
(8)Arkeryd, L. Catastrophe theory in Hilbert space. Preprint of the Chalmers Univ. of Technology and the Univ. of Göteborg, no. 1977–17.Google Scholar