Obstruction theory and multiparameter Hopf bifurcation

Ize, Jorge

doi:10.1090/S0002-9947-1985-0784013-2

Obstruction theory and multiparameter Hopf bifurcation
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by Jorge Ize

Trans. Amer. Math. Soc. 289 (1985), 757-792

DOI: https://doi.org/10.1090/S0002-9947-1985-0784013-2

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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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