Centre Manifolds, Normal Forms and Elementary Bifurcations
Centre Manifolds, Normal Forms and Elementary Bifurcations
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DOI：
10.1007/978-3-322-96657-5_4
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发表时间：
1989
期刊：
影响因子：
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通讯作者：
A. Vanderbauwhede
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文献类型：
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作者：
A. Vanderbauwhede
这些说明起源于一个研讨会上举行的动力系统在大学的Louvain-la-Neuve（比利时）在1985年春天。我们的指南，研讨会是图书非线性振荡，动力系统和分叉的向量场由Guckenheimer和霍姆斯[9]。这本优秀的书有一个缺点，数学家：它包含很少的证明，因此，一个被迫去搜索的文献，如果一个人想填补的细节。当我试图这样做的第3章的书（中心流形，规范形式理论，余维一分叉），我很快就感到沮丧，而粗略的方式，其中大多数文本处理更技术的部分，这一理论。大约在同一时间，我在S.货车Gils [36]的想法，使用空间的指数增长的功能，以制定和证明中心流形定理。这似乎（至少对我来说）是一个更自然的方法，后来我们发现其他人也已经在某种程度上使用了这个想法。在几位同事的激励下，我开始着手写一些笔记，这些笔记将包含对该理论的合理完整的描述，将使用新方法作为其主要指导原则，并且适合在研讨会和研究生课程中使用。当然，我严重低估了完成这样一个项目所需的努力，直到1986年夏天，我才能够制作出第一个版本。从那时起，这一版本在一些同事中流传，并在几次研讨会上试用。许多意见和建议，我收到的考虑时，我写的版本在这里提出的主要区别与早期版本是使用的纤维收缩定理证明可微的中心流形。我把它留给读者来判断文本在多大程度上仍然反映了它最初的目标。
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 bookNonlinear Oscillations,Dynamical Systems and Bifurcations of Vector Fieldsby 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.