Qualitative Theory of Planar Differential Systems

  • Freddy Dumortier
  • Jaume Llibre
  • Joan C. Artés

About this book

Introduction

The book deals essentially with systems of polynomial autonomous ordinary differential equations in two real variables. The emphasis is mainly qualitative, although attention is also given to more algebraic aspects as a thorough study of the center/focus problem and recent results on integrability. In the last two chapters the performant software tool P4 is introduced: based on both algebraic manipulation and numerical calculation, this was conceived for the purpose of drawing "Polynomial Planar Phase Portraits" on part of the plane, or on a Poincaré compactification, or even on a Poincaré-Lyapunov compactification of the plane.

From the start, differential systems are represented by vector fields enabling, in full strength, a dynamical systems approach. All essential notions, including invariant manifolds, normal forms, desingularization of singularities, index theory and limit cycles, are introduced and the main results are proved for smooth systems with the necessary specifications for analytic and polynomial systems.

The book is very appropriate for a first course in dynamical systems, presenting the basic notions in the study of individual two dimensional systems. Not only does it provide simple and appropriate proofs, but it also contains a lot of exercises and presents a survey of interesting results with the necessary references to the literature.

Keywords

Dynamical systems ODE ordinary differential equation phase portrait planar differential systems qualitative theory

Authors and affiliations

  • Freddy Dumortier
    • 1
  • Jaume Llibre
    • 2
  • Joan C. Artés
    • 3
  1. 1.Hasselt UniversityDiepenbeekBelgium
  2. 2.Dept. MatemátiquesUniversitat Autònoma deBarcelonaSpain
  3. 3.Dept. MatemátiquesUniversitat Autònoma deBarcelonaSpain

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-540-32902-2
  • Copyright Information Springer 2006
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-540-32893-3
  • Online ISBN 978-3-540-32902-2
  • About this book