Geometric theory of dynamical systems
Geometric theory of dynamical systems
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DOI：
10.1007/978-1-4612-5703-5
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发表时间：
1982
期刊：
影响因子：
--
通讯作者：
J. Palis;W. Melo
中科院分区：
文献类型：
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作者：
J. Palis;W. Melo
... 定性研究（desequationsdifj'erentielles）aura par elle-m me un inter t du prime ordre... HENRI POINCARE，1881。我们在本书中提出了动力系统几何理论的观点，它是介绍性的，但让读者了解涉及两个重要主题的一些基本思想：结构稳定性和通用性。从庞加莱、李雅普诺夫和伯克霍夫开始，许多数学家都考虑过这个理论。近年来，它确立了一些总体目标，并经历了长足的发展。两件重要事件之间已经过去了二十多年：Andronov 和 Pontryagin (1937) 的工作介绍了结构稳定性的基本概念，而 Peixoto (1958-1962) 的文章证明了表面上稳定矢量场的密度。就在那时，斯梅尔将寻找通用和稳定的性质作为主要目标，并通过获得结果并提出与此背景密切相关的问题，极大地丰富了该理论。在同一时期，哈特曼和格罗布曼表明，局部稳定性是一种通用属性。不久之后，库普卡和斯梅尔成功解决了周期轨道问题。我们打算通过许多例子以及 Hartman-Grobman 和稳定流形定理（第 2 章）、Kupka-Smale 定理（第 3 章）和 Peixoto 定理（第 4 章）的系统证明，让读者了解这一理论。我们给出的几个证明比原来的证明更简单，并且可以进行重要的概括。
... cette etude qualitative (des equations difj'erentielles) aura par elle-m me un inter t du premier ordre... HENRI POINCARE, 1881. We present in this book a view of the Geometric Theory of Dynamical Systems, which is introductory and yet gives the reader an understanding of some of the basic ideas involved in two important topics: structural stability and genericity. This theory has been considered by many mathematicians starting with Poincare, Liapunov and Birkhoff. In recent years some of its general aims were established and it experienced considerable development. More than two decades passed between two important events: the work of Andronov and Pontryagin (1937) introducing the basic concept of structural stability and the articles of Peixoto (1958-1962) proving the density of stable vector fields on surfaces. It was then that Smale enriched the theory substantially by defining as a main objective the search for generic and stable properties and by obtaining results and proposing problems of great relevance in this context. In this same period Hartman and Grobman showed that local stability is a generic property. Soon after this Kupka and Smale successfully attacked the problem for periodic orbits. We intend to give the reader the flavour of this theory by means of many examples and by the systematic proof of the Hartman-Grobman and the Stable Manifold Theorems (Chapter 2), the Kupka-Smale Theorem (Chapter 3) and Peixoto's Theorem (Chapter 4). Several ofthe proofs we give vii Introduction Vlll are simpler than the original ones and are open to important generalizations.