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Applications of the blowing-up construction and algebraic geometry to bifurcation problems

Buchner, Michael and Marsden, Jerrold and Schecter, Stephen (1983) Applications of the blowing-up construction and algebraic geometry to bifurcation problems. Journal of Differential Equations, 48 (3). pp. 404-433. ISSN 0022-0396. doi:10.1016/0022-0396(83)90102-X.

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A generalization of the Morse lemma to vector-valued functions is proved by a blowing-up argument. This is combined with a theorem from algebraic geometry on the number of real solutions of a system of homogeneous equations of even degree to yield a new bifurcation theorem. Bifurcation in a one- or multi-parameter problem is guaranteed if the leading term is of even degree (it is often two) and satisfies a regularity condition. Applications are given to nonlinear eigenvalue problems and to the Hopf bifurcation.

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Additional Information:© 1983 Published by Elsevier Inc. Received 14 September 1981; Revised 22 January 1982. Available online 7 September 2004. A number or people have supplied us with valuable comments and preprints which were userul in the development or our results. Amons these are Judy Arms. David Chillingworth. Shu·Nee Chow. E. N. Dancer. Mike Fitzpatrick. Marty Golubitsky, John Guckenheimer. Jack Hale. Robert Magnus, David SchaelTer, Michael Shearer, Michael Singer, Alan Weinstein. and Jim Yorke.
Issue or Number:3
Record Number:CaltechAUTHORS:20100715-114630209
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ID Code:19079
Deposited By: Ruth Sustaita
Deposited On:15 Jul 2010 18:59
Last Modified:08 Nov 2021 23:49

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