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Über dieses Buch

These notes are an elaboration of the first part of a course on foliations which I have given at Strasbourg in 1976 and at Tunis in 1977. They are concerned mostly with dynamical sys­ tems in dimensions one and two, in particular with a view to their applications to foliated manifolds. An important chapter, however, is missing, which would have been dealing with structural stability. The publication of the French edition was re­ alized by-the efforts of the secretariat and the printing office of the Department of Mathematics of Strasbourg. I am deeply grateful to all those who contributed, in particular to Mme. Lambert for typing the manuscript, and to Messrs. Bodo and Christ for its reproduction. Strasbourg, January 1979. Table of Contents I. VECTOR FIELDS ON MANIFOLDS 1. Integration of vector fields. 1 2. General theory of orbits. 13 3. Irlvariant and minimaI sets. 18 4. Limit sets. 21 5. Direction fields. 27 A. Vector fields and isotopies. 34 II. THE LOCAL BEHAVIOUR OF VECTOR FIELDS 39 1. Stability and conjugation. 39 2. Linear differential equations. 44 3. Linear differential equations with constant coefficients. 47 4. Linear differential equations with periodic coefficients. 50 5. Variation field of a vector field. 52 6. Behaviour near a singular point. 57 7. Behaviour near a periodic orbit. 59 A. Conjugation of contractions in R. 67 III. PLANAR VECTOR FIELDS 75 1. Limit sets in the plane. 75 2. Periodic orbits. 82 3. Singular points. 90 4. The Poincare index.

Inhaltsverzeichnis

Frontmatter

Chapter I. Vector Fields on Manifolds

Abstract
Let M be a differentiable manifold without boundary of dimension m and of class Cs, 2 ⩽ s ⩽ + ∞ (respectively analytic), and let X be a vector field on M of class cr, 1 ⩽ r ⩽ s−1 (respectively analytic).
Claude Godbillon

Chapter II. The Local Behaviour of Vector Fields

Abstract
In this chapter we shall present some properties of linear differential equations in ℝm in order to study locally the orbits of a vector field X on a manifold M in a neighbourhood of a singular point or of a periodic orbit.
Claude Godbillon

Chapter III. Planar Vector Fields

Abstract
The investigation of vector fields on open sets of the plane was developed by H. Poincaré and given a more precise form by I. Bendixson; it is nowadays known as the Poincaré-Bendixson theory. It is based in an essential way on the following version of Jordan’s theorem (cf. [16]):
Claude Godbillon

Chapter IV. Direction Fields on the Torus and Homeomorphisms of the Circle

Abstract
The first results on this subject are again due to H.Poincaré who showed in particular the rôle of the homeomorphisms of the circle in this context. He discovered the phenomenon of the exceptional homeomorphisms which was further clarified later on by A. Denjoy, as well as the properties of conjugation to rotations which recently were continued by V. Arnold and M. Herman. Furthermore the global qualitative description of direction fields on the torus and on the Klein bottle are due to H. Kneser.
Claude Godbillon

Chapter V. Vector Fields on Surfaces

Without Abstract
Claude Godbillon

Backmatter

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