The transformation of vector-functions, scaling and bifurcation
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- by R. J. Magnus PDF
- Trans. Amer. Math. Soc. 286 (1984), 689-713 Request permission
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.References
- Michael Buchner, Jerrold Marsden, and Stephen Schecter, Applications of the blowing-up construction and algebraic geometry to bifurcation problems, J. Differential Equations 48 (1983), no. 3, 404–433. MR 702428, DOI 10.1016/0022-0396(83)90102-X
- Shui Nee Chow, Jack K. Hale, and John Mallet-Paret, Applications of generic bifurcation. II, Arch. Rational Mech. Anal. 62 (1976), no. 3, 209–235. MR 415673, DOI 10.1007/BF00280015
- Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI 10.1016/0022-1236(71)90015-2
- Nicolaas H. Kuiper, $C^{1}$-equivalence of functions near isolated critical points, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967) Ann. of Math. Studies, No. 69, Princeton Univ. Press, Princeton, N. J., 1972, pp. 199–218. MR 0413161
- Tzee Char Kuo, Characterizations of $v$-sufficiency of jets, Topology 11 (1972), 115–131. MR 288775, DOI 10.1016/0040-9383(72)90026-2
- R. J. Magnus, On the local structure of the zero-set of a Banach space valued mapping, J. Functional Analysis 22 (1976), no. 1, 58–72. MR 0418149, DOI 10.1016/0022-1236(76)90023-9 —, The reduction of a vector-valued function near a critical point, Battelle-Geneva Math. Report no. 93, 1975.
- Robert Magnus, Topological equivalence in bifurcation theory, Functional differential equations and bifurcation (Proc. Conf., Inst. Ciênc. Mat. São Carlos, Univ. São Paulo, São Carlos, 1979) Lecture Notes in Math., vol. 799, Springer, Berlin, 1980, pp. 263–276. MR 585491
- Jerrold E. Marsden, Qualitative methods in bifurcation theory, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1125–1148. MR 508450, DOI 10.1090/S0002-9904-1978-14549-2
- M. Shearer, Bifurcation in the neighbourhood of a non-isolated singular point, Israel J. Math. 30 (1978), no. 4, 363–381. MR 501095, DOI 10.1007/BF02762000
- Ivan Kupka, Contribution à la théorie des champs génériques, Contributions to Differential Equations 2 (1963), 457–484 (French). MR 165536
- Andrzej Szulkin, Local structure of the zero-sets of differentiable mappings and application to bifurcation theory, Math. Scand. 45 (1979), no. 2, 232–242. MR 580601, DOI 10.7146/math.scand.a-11838
- C. A. Stuart, Bifurcation for variational problems when the linearisation has no eigenvalues, J. Functional Analysis 38 (1980), no. 2, 169–187. MR 587907, DOI 10.1016/0022-1236(80)90063-4
- J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
Additional Information
- © Copyright 1984 American Mathematical Society
- 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