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Obstruction theory and multiparameter Hopf bifurcation


Author: Jorge Ize
Journal: Trans. Amer. Math. Soc. 289 (1985), 757-792
MSC: Primary 58E07; Secondary 55S35, 58F22
DOI: https://doi.org/10.1090/S0002-9947-1985-0784013-2
MathSciNet review: 784013
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0784013-2
Keywords: Hopf bifurcation, equivariant obstruction
Article copyright: © Copyright 1985 American Mathematical Society

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