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The transformation of vector-functions, scaling and bifurcation


Author: R. J. Magnus
Journal: Trans. Amer. Math. Soc. 286 (1984), 689-713
MSC: Primary 58E07; Secondary 47H15
DOI: https://doi.org/10.1090/S0002-9947-1984-0760981-9
MathSciNet review: 760981
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Abstract: Various known methods for studying the bifurcation of zeros of a Banach-space-valued 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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  • [1] M. Buchner, J. Marsden and S. Schecter, Applications of the blowing-up construction and algebraic geometry to bifurcation problems, Preprint, Univ. of California, Berkeley, 1981. MR 702428 (84m:58027)
  • [2] S-N. Chow, J. K. Hale and J. Mallet-Paret, Applications of generic bifurcation. II, Arch. Rational Mech. Anal. 62 (1976), 209-325. MR 0415673 (54:3753)
  • [3] M. G. Crandall and P. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321-340. MR 0288640 (44:5836)
  • [4] N. H. Kuiper, $ {C^1}$-equivalence of functions near isolated critical points, Symposium on Infinite-Dimensional Topology, edited by R. D. Anderson, Princeton Univ. Press, Princeton, N.J.; Univ. of Tokyo Press, Tokyo, 1972. MR 0413161 (54:1282)
  • [5] T-C. Kuo, Characterisation of $ v$-sufficiency of jets, Topology 11 (1972), 115-131. MR 0288775 (44:5971)
  • [6] R. J. Magnus, On the local structure of the zero-set of a Banach space valued mapping, J. Funct. Anal. 22 (1976), 58-72. MR 0418149 (54:6191)
  • [7] -, The reduction of a vector-valued function near a critical point, Battelle-Geneva 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. 263-276. MR 585491 (81m:58020)
  • [9] J. Marsden, Qualitative methods in bifurcation theory Bull. Amer. Math. Soc. 84 (1978), 1125-1148. MR 508450 (80a:58012)
  • [10] M. Shearer, Bifurcation in the neighbourhood of a non-isolated critical point, Israel J. Math. 30 (1978), 363-381. MR 0501095 (58:18547)
  • [11] S. Smale, Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963), 97-116. MR 0165537 (29:2818b)
  • [12] A. Szulkin, Local structure of the zero-sets of differentiable mappings and application to bifurcation theory, Math. Scand. 45 (1979), 232-242. MR 580601 (82d:58014)
  • [13] C. A. Stuart, Bifurcation for variational problems when the linearisation has no eigenvalues, J. Funct. Anal. 38 (1980), 169-187. 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: https://doi.org/10.1090/S0002-9947-1984-0760981-9
Article copyright: © Copyright 1984 American Mathematical Society

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