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Centre manifolds, normal forms and elementary bifurcations. (English) Zbl 0677.58001

Dynamics reported, Vol. 2, 89-169 (1989).
[For the entire collection see Zbl 0659.00009.]
The paper presents first a detailed proof of the centre manifold theorem for flows (existence, differentiability and questions of uniqueness). Then normal forms of vector fields near singular points are discussed, and the last part concerns codimension 1 bifurcations, i.e. bifurcations which may appear generically in the flows belonging to one parameter families of vector fields. It is the author’s (attained) aim to complete chapter 3 of the book “Nonlinear oscillations, dynamical systems and bifurcation theory” by J. Guckenheimer and P. Holmes [Appl. Math. Sci. 42 (1983; Zbl 0515.34001)] where technical details of the proofs are omitted. The proof of the centre manifold theorem is inspired by ideas of S. A. van Gils [see his common paper with the author in J. Funct. Anal. 72, 209-224 (1987; Zbl 0621.47050)] and uses a scale of Banach spaces consisting of exponentially growing functions. Bibliographical notes give references to papers which appeared after the book of Guckenheimer and Holmes was edited.
Reviewer: H.G.Bothe

MSC:

58-02 Research exposition (monographs, survey articles) pertaining to global analysis
37G99 Local and nonlocal bifurcation theory for dynamical systems