The transformation of vector-functions, scaling and bifurcation

Magnus, R. J.

doi:10.1090/S0002-9947-1984-0760981-9

The transformation of vector-functions, scaling and bifurcation
HTML articles powered by AMS MathViewer

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

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