Geometric Theory of Dynamical Systems: An Introduction

ISBN-13: 9781461257059 / Angielski / Miękka / 2012 / 198 str.

J. Jr. Palis; W. De Melo; J. Jr. Palis

Opis

... 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.

Spis treści

1 Differentiable Manifolds and Vector Fields.- §0 Calculus in ?n and Differentiable Manifolds.- §1 Vector Fields on Manifolds.- §2 The Topology of the Space of Cr Maps.- §3 Transversality.- §4 Structural Stability.- 2 Local Stability.- §1 The Tubular Flow Theorem.- §2 Linear Vector Fields.- §3 Singularities and Hyperbolic Fixed Points.- §4 Local Stability.- §5 Local Classification.- §6 Invariant Manifolds.- §7 The ?-lemma (Inclination Lemma). Geometrical Proof of Local Stability.- 3 The Kupka-Smale Theorem.- §1 The Poincaré Map.- §2 Genericity of Vector Fields Whose Closed Orbits Are Hyperbolic.- §3 Transversality of the Invariant Manifolds.- 4 Genericity and Stability of Morse-Smale Vector Fields.- §1 Morse-Smale Vector Fields; Structural Stability.- §2 Density of Morse-Smale Vector Fields on Orientable Surfaces.- §3 Generalizations.- §4 General Comments on Structural Stability. Other Topics.- Appendix: Rotation Number and Cherry Flows.- References.