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© 2000

Bifurcations and Catastrophes

Geometry of Solutions to Nonlinear Problems

Textbook

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages I-VIII
  2. Michel Demazure
    Pages 1-11
  3. Michel Demazure
    Pages 13-30
  4. Michel Demazure
    Pages 31-61
  5. Michel Demazure
    Pages 63-86
  6. Michel Demazure
    Pages 87-113
  7. Michel Demazure
    Pages 115-145
  8. Michel Demazure
    Pages 147-177
  9. Michel Demazure
    Pages 179-217
  10. Michel Demazure
    Pages 219-247
  11. Michel Demazure
    Pages 249-268
  12. Michel Demazure
    Pages 269-292
  13. Back Matter
    Pages 293-303

About this book

Introduction

Based on a lecture course at the Ecole Polytechnique (Paris), this text gives a rigorous introduction to many of the key ideas in nonlinear analysis, dynamical systems and bifurcation theory including catastrophe theory. Wherever appropriate it emphasizes a geometrical or coordinate-free approach which allows a clear focus on the essential mathematical structures. Taking a unified view, it brings out features common to different branches of the subject while giving ample references for more advanced or technical developments.

Keywords

Bifurcations Catastrophes Dynamical Systems Maxima Nonlinear Singularities catastrophe theory diffeomorphism manifold

Authors and affiliations

  1. 1.Cité des Sciences et de l’IndustrieParisFrance

Bibliographic information

  • Book Title Bifurcations and Catastrophes
  • Book Subtitle Geometry of Solutions to Nonlinear Problems
  • Authors Michel Demazure
  • Series Title Universitext
  • DOI https://doi.org/10.1007/978-3-642-57134-3
  • Copyright Information Springer-Verlag Berlin Heidelberg 2000
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Softcover ISBN 978-3-540-52118-1
  • eBook ISBN 978-3-642-57134-3
  • Series ISSN 0172-5939
  • Series E-ISSN 2191-6675
  • Edition Number 1
  • Number of Pages VIII, 304
  • Number of Illustrations 1 b/w illustrations, 0 illustrations in colour
  • Additional Information Original French edition published by Ellipses, 1998
  • Topics Differential Geometry
    Global Analysis and Analysis on Manifolds
    Dynamical Systems and Ergodic Theory
  • Buy this book on publisher's site

Reviews

"This book gives an introduction to the basic ideas in dynamical systems and catastrophe and bifurcation theory. It starts with the geometrical concepts which are necessary for the rest of the book. In the first four chapters, the author introduces the notion of local inversion for maps, submanifolds, tranversality, and the classical theorems related to the local theory of critical points, that is, Sard's theorem and Morse's lemma. After a study of the classification of differentiable maps, he introduces the notion of germ and shows how catastrophe theory can be used to classify singularities; elementary catastrophes are discussed in Chapter 5. Vector fields are the subject of the rest of the book. Chapter 6 is devoted to enunciating the existence and uniqueness theorems for ordinary differential equations; the notions of first integral, one-parameter group and phase portrait are also introduced in this part. Linear vector fields and the topological classification of flows are studied in Chapter 7. Chapter 8 is devoted to the classification of singular points of vector fields. Lyapunov theory and the theorems of Grobman and Hartman are also described in this chapter. The notions of Poincaré map and closed orbit, and the concepts necessary for the classification of closed orbits, are the principal ideas of Chapter 9; this chapter finishes with the notion of structural stability and the classification of structurally stable vector fields in dimension 2 and Morse-Smale vector fields. Finally, in Chapter 10 the author defines the idea of bifurcation of phase portraits and describes the simplest local bifurcations: saddle-node bifurcation, Hopf bifurcation, etc.

This book can be used as a textbook for a first course on dynamical systems and bifurcation theory."  (Joan Torregrosa, Mathematical Reviews)