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Nonlinear Differential Equations and Dynamical Systems

  • Textbook
  • © 1996

Overview

Authors:
  1. Ferdinand Verhulst

    1. Department of Mathematics, University of Utrecht, Utrecht, The Netherlands

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Part of the book series: Universitext (UTX)

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Table of contents (15 chapters)

  1. Front Matter

    Pages I-X
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  2. Introduction

    • Ferdinand Verhulst
    Pages 1-6

  3. Autonomous equations

    • Ferdinand Verhulst
    Pages 7-24

  4. Critical points

    • Ferdinand Verhulst
    Pages 25-37

  5. Periodic solutions

    • Ferdinand Verhulst
    Pages 38-58

  6. Introduction to the theory of stability

    • Ferdinand Verhulst
    Pages 59-68

  7. Linear Equations

    • Ferdinand Verhulst
    Pages 69-82

  8. Stability by linearisation

    • Ferdinand Verhulst
    Pages 83-95

  9. Stability analysis by the direct method

    • Ferdinand Verhulst
    Pages 96-109

  10. Introduction to perturbation theory

    • Ferdinand Verhulst
    Pages 110-121

  11. The Poincaré-Lindstedt method

    • Ferdinand Verhulst
    Pages 122-135

  12. The method of averaging

    • Ferdinand Verhulst
    Pages 136-165

  13. Relaxation Oscillations

    • Ferdinand Verhulst
    Pages 166-172

  14. Bifurcation Theory

    • Ferdinand Verhulst
    Pages 173-192

  15. Chaos

    • Ferdinand Verhulst
    Pages 193-223

  16. Hamiltonian systems

    • Ferdinand Verhulst
    Pages 224-247

  17. Back Matter

    Pages 248-305
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Keywords

About this book

On the subject of differential equations many elementary books have been written. This book bridges the gap between elementary courses and research literature. The basic concepts necessary to study differential equations - critical points and equilibrium, periodic solutions, invariant sets and invariant manifolds - are discussed first. Stability theory is then developed starting with linearisation methods going back to Lyapunov and Poincaré. In the last four chapters more advanced topics like relaxation oscillations, bifurcation theory, chaos in mappings and differential equations, Hamiltonian systems are introduced, leading up to the frontiers of current research: thus the reader can start to work on open research problems, after studying this book. This new edition contains an extensive analysis of fractal sets with dynamical aspects like the correlation- and information dimension. In Hamiltonian systems, topics like Birkhoff normal forms and the Poincaré-Birkhoff theorem on periodicsolutions have been added. There are now 6 appendices with new material on invariant manifolds, bifurcation of strongly nonlinear self-excited systems and normal forms of Hamiltonian systems. The subject material is presented from both the qualitative and the quantitative point of view, and is illustrated by many examples.

Authors and Affiliations

  • Department of Mathematics, University of Utrecht, Utrecht, The Netherlands

    Ferdinand Verhulst

About the author

Ferdinand Verhulst was born in Amsterdam, The Netherlands, in 1939.

He graduated at the University of Amsterdam in Astrophysics and Mathematics. A period of five years at the Technological University of Delft, started his interest in technological problems, resulting in various cooperations with engineers. His other interests include the methods and applications of asymptotic analysis, nonlinear oscillations and wave theory.

He holds a chair of dynamical systems at the department of mathematics at the University of Utrecht.

Among his other interests are a publishing company, Epsilon Uitgaven, that he founded in 1985, and the relation between dynamical systems and psychoanalysis.

For more information see www.math.uu.nl/people/verhulst

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