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© 1998

Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem

Book

Part of the Modern Birkhäuser Classics book series (MBC)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Robert Roussarie
    Pages 1-15
  3. Robert Roussarie
    Pages 17-31
  4. Robert Roussarie
    Pages 33-49
  5. Robert Roussarie
    Pages 51-90
  6. Robert Roussarie
    Pages 91-149
  7. Back Matter
    Pages 193-204

About this book

Introduction

In a coherent, exhaustive and progressive way, this book presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. A systematic introduction is given to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation. The exposition moves from classical analytic geometric methods applied to regular limit periodic sets to more recent tools for singular limit sets.

The methods can be applied to theoretical problems such as Hilbert's 16th problem, but also for the purpose of establishing bifurcation diagrams of specific families as well as explicit computations. 

- - -

The book as a whole is a well-balanced exposition that can be recommended to all those who want to gain a thorough understanding and proficiency in recently developed methods. The book, reflecting the state of the art, can also be used for teaching special courses.
(Mathematical Reviews)

 

Keywords

bifurcation diagrams desingularization theory differential equations dynamical systems limit cycles limit periodic sets singular limit sets bifurcation differential equation

Authors and affiliations

  1. 1.Department of MathematicsUniversity of BourgogneDijon CedexFrance

About the authors

Robert Roussarie was Professor of Mathematics at the University of Burgundy, France.       

Bibliographic information

  • Book Title Bifurcations of Planar Vector Fields and Hilbert's Sixteenth Problem
  • Authors Robert Roussarie
  • Series Title Modern Birkhäuser Classics
  • Series Abbreviated Title Modern Birkhäuser Classics
  • DOI https://doi.org/10.1007/978-3-0348-0718-0
  • Copyright Information Springer Basel 1998
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-3-7643-5900-3
  • Softcover ISBN 978-3-0348-9778-5
  • eBook ISBN 978-3-0348-0718-0
  • Series ISSN 2197-1803
  • Series E-ISSN 2197-1811
  • Edition Number 1
  • Number of Pages XVII, 206
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Additional Information Originally published as volume 164 in the Progress in Mathematics series
  • Topics Analysis
    Global Analysis and Analysis on Manifolds
    Partial Differential Equations
    Dynamical Systems and Ergodic Theory
  • Buy this book on publisher's site

Reviews

"A well-balanced exposition that can be recommended to all those who want to gain a thorough understanding and proficiency in recently developed methods. The book, reflecting the state of the art, can also be used for teaching special courses."

--Mathematical Reviews