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Stability of universal unfoldings: a correction

Published online by Cambridge University Press:  24 October 2008

R. J. Magnus
R. J. Magnus
Affiliation:
University of Iceland, Reykjavik
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Extract

The author is grateful to Les Lander for pointing out an error in the stability section of (1). In fact Theorems 5 and 7 are incorrect. Recently Arkeryd proved a stability theorem for the infinite-dimensional case in the context of the imperfect bifurcation theory of Golubitsky and Schaeffer(3). In his result finitely many derivatives are controlled, the number depending on the codimension of the singularity unfolded. In this note we shall present a stability theorem involving the determinacy of the singularity. The context is the parameter-free potential case, that is, catastrophe theory. The proof is without recourse to the finite-dimensional results, and the theorem concludes an account of a part of singularity theory in Banach spaces, in which the author has tried to use as little as possible of the finite-dimensional theory (1, 2).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 1 , July 1981 , pp. 195 - 196
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

REFERENCES

(1)Magnus, R. J.Universal unfoldings in Banach spaces: reduction and stability. Math. Proc. Cambridge Philos. Soc. 86, (1979), 41–55.CrossRefGoogle Scholar
(2)Magnus, R. J.Determinacy in a class of germs on a reflexive Banach space. Math. Proc. Cambridge Philos Soc. 84 (1978), 293–302.CrossRefGoogle Scholar
(3)Arkeryd, L. Imperfect bifurcation and Banach space singularity theory (Preprint Chalmers University of Technology and the University of Göteborg).Google Scholar
(4)Trotman, D. J. A. and Zeeman, E. C. The classification of elementary catastrophes of codimension ≤ 5. In Structural Stability, the Theory of Catastrophes, and Applications in the Sciences, Lecture Notes in Mathematics vol. 525. ed. Hilton, P. J. (Springer, Berlin and New York, 1976) pp. 263–327.Google Scholar