The transformation of vectorfunctions, scaling and bifurcation
Author:
R. J. Magnus
Journal:
Trans. Amer. Math. Soc. 286 (1984), 689713
MSC:
Primary 58E07; Secondary 47H15
MathSciNet review:
760981
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Abstract: Various known methods for studying the bifurcation of zeros of a Banachspacevalued mapping are unified under a single idea, akin to using a coordinate transformation to obtain a simple form of the function under consideration. The general nature of the hypotheses permits the dropping of the pervasive "Fredholm condition" of bifurcation theory.
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 M. Buchner, J. Marsden and S. Schecter, Applications of the blowingup construction and algebraic geometry to bifurcation problems, Preprint, Univ. of California, Berkeley, 1981. MR 702428 (84m:58027)
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 M. G. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321340. MR 0288640 (44:5836)
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 N. H. Kuiper, equivalence of functions near isolated critical points, Symposium on InfiniteDimensional Topology, edited by R. D. Anderson, Princeton Univ. Press, Princeton, N.J.; Univ. of Tokyo Press, Tokyo, 1972. MR 0413161 (54:1282)
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 TC. Kuo, Characterisation of sufficiency of jets, Topology 11 (1972), 115131. MR 0288775 (44:5971)
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 R. J. Magnus, On the local structure of the zeroset of a Banach space valued mapping, J. Funct. Anal. 22 (1976), 5872. MR 0418149 (54:6191)
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 , The reduction of a vectorvalued function near a critical point, BattelleGeneva Math. Report no. 93, 1975.
 [8]
 , Topological equivalence in bifurcation theory, Functional Differential Equations and Bifurcation, Lecture Notes in Math., vol. 799, Springer, Berlin and New York, pp. 263276. MR 585491 (81m:58020)
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 J. Marsden, Qualitative methods in bifurcation theory Bull. Amer. Math. Soc. 84 (1978), 11251148. MR 508450 (80a:58012)
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 M. Shearer, Bifurcation in the neighbourhood of a nonisolated critical point, Israel J. Math. 30 (1978), 363381. MR 0501095 (58:18547)
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 S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 97116. MR 0165537 (29:2818b)
 [12]
 A. Szulkin, Local structure of the zerosets of differentiable mappings and application to bifurcation theory, Math. Scand. 45 (1979), 232242. MR 580601 (82d:58014)
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 C. A. Stuart, Bifurcation for variational problems when the linearisation has no eigenvalues, J. Funct. Anal. 38 (1980), 169187. MR 587907 (82c:58015)
 [14]
 J. Dieudonné, Foundations of modern analysis, Academic Press, New York and London, 1960. MR 0120319 (22:11074)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198407609819
PII:
S 00029947(1984)07609819
Article copyright:
© Copyright 1984 American Mathematical Society
