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© 1990

Nonlinear Differential Equations and Dynamical Systems

Textbook

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages I-IX
  2. Ferdinand Verhulst
    Pages 1-6
  3. Ferdinand Verhulst
    Pages 7-26
  4. Ferdinand Verhulst
    Pages 27-38
  5. Ferdinand Verhulst
    Pages 39-61
  6. Ferdinand Verhulst
    Pages 62-72
  7. Ferdinand Verhulst
    Pages 73-87
  8. Ferdinand Verhulst
    Pages 88-100
  9. Ferdinand Verhulst
    Pages 101-116
  10. Ferdinand Verhulst
    Pages 117-129
  11. Ferdinand Verhulst
    Pages 130-144
  12. Ferdinand Verhulst
    Pages 145-176
  13. Ferdinand Verhulst
    Pages 177-182
  14. Ferdinand Verhulst
    Pages 183-203
  15. Ferdinand Verhulst
    Pages 204-217
  16. Ferdinand Verhulst
    Pages 218-236
  17. Back Matter
    Pages 237-280

About this book

Introduction

On the subject of differential equations a great many elementary books have been written. This book bridges the gap between elementary courses and the research literature. The basic concepts necessary to study differential equations - critical points and equilibrium, periodic solutions, invariant sets and invariant manifolds - are discussed. Stability theory is developed starting with linearisation methods going back to Lyapunov and Poincaré. The global direct method is then discussed. To obtain more quantitative information the Poincaré-Lindstedt method is introduced to approximate periodic solutions while at the same time proving existence by the implicit function theorem. The method of averaging is introduced as a general approximation-normalisation method. The last four chapters introduce the reader to relaxation oscillations, bifurcation theory, centre manifolds, chaos in mappings and differential equations, Hamiltonian systems (recurrence, invariant tori, periodic solutions). The book presents the subject material from both the qualitative and the quantitative point of view. There are many examples to illustrate the theory and the reader should be able to start doing research after studying this book.

Keywords

Implicit function averaging methods bifurcation theory chaos differential equation differential equations dynamical systems dynamische Systeme nonlinear systems oscillations

Authors and affiliations

  1. 1.Department of MathematicsUniversity of UtrechtTA UtrechtThe Netherlands

About the authors

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

Bibliographic information

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