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Obstruction Theory and Multiparameter Hopf Bifurcation

Jorge Ize
Transactions of the American Mathematical Society
Vol. 289, No. 2 (Jun., 1985), pp. 757-792
DOI: 10.2307/2000262
Stable URL: http://www.jstor.org/stable/2000262
Page Count: 36
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Obstruction Theory and Multiparameter Hopf Bifurcation
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Abstract

The Hopf bifurcation problem is treated as an example of an equivariant bifurcation. The existence of a local bifurcating solution is given by the nonvanishing of an obstruction to extending a map defined on a complex projective space and is computed using the complex Bott periodicity theorem. In the case of the classical Hopf bifurcation the results of Chow, Mallet-Paret and Yorke are recovered without using any special index as the Fuller degree: There is bifurcation if the number of exchanges of stability is nonzero. A global theorem asserts that the sum of the local invariants on a bounded component of solutions must be zero.

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