Abstract
These notes originated from a seminar on dynamical systems held at the University of Louvain-la-Neuve (Belgium) in the spring of 1985. Our guide for that seminar was the book Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields by Guckenheimer and Holmes [9]. This otherwise excellent book has one disadvantage for mathematicians: it contains very few proofs, and hence one is forced to go searching in the literature if one wants to fill in the details. When I tried to do this for chapter 3 of the book (on centre manifolds, normal form theory, and codimension one bifurcations), I rapidly got frustrated by the rather sketchy way in which most texts deal with the more technical parts of this theory. Around the same time I found in the PhD thesis of S. Van Gils [36] the idea of using spaces of exponentially growing functions in order to formulate and prove the centre manifold theorem. This seemed (at least for me) to be a more natural approach, and later we found out that also others had already used this idea to some extent. Stimulated by a few colleagues I then started on the project of writing some notes which would contain a reasonably complete account of the theory, which would use the new approach as its main guiding principle, and which would be suitable for use in seminars and graduate courses. Of course I had seriously underestimated the efforts needed to finish such a project, and it was only with a considerable delay that I was able to produce a first version during the summer of 1986. Since then this version has circulated among some colleagues and was tried out at a few seminars. The many remarks and suggestions which I received were taken into consideration when I wrote the version presented here; the main difference with the earlier version is the use of the fibre contraction theorem to prove the differentiability of the centre manifold. I leave it to the reader to judge to what extent the text still reflects its original goals.
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Vanderbauwhede, A. (1989). Centre Manifolds, Normal Forms and Elementary Bifurcations. In: Kirchgraber, U., Walther, H.O. (eds) Dynamics Reported. Dynamics Reported, vol 2. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-322-96657-5_4
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