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Asymptotic Methods for Relaxation Oscillations and Applications

  • Johan Grasman

Part of the Applied Mathematical Sciences book series (AMS, volume 63)

Table of contents

  1. Front Matter
    Pages N2-xiii
  2. Johan Grasman
    Pages 1-22
  3. Johan Grasman
    Pages 23-114
  4. Johan Grasman
    Pages 115-150
  5. Back Matter
    Pages 201-223

About this book

Introduction

In various fields of science, notably in physics and biology, one is con­ fronted with periodic phenomena having a remarkable temporal structure: it is as if certain systems are periodically reset in an initial state. A paper of Van der Pol in the Philosophical Magazine of 1926 started up the investigation of this highly nonlinear type of oscillation for which Van der Pol coined the name "relaxation oscillation". The study of relaxation oscillations requires a mathematical analysis which differs strongly from the well-known theory of almost linear oscillations. In this monograph the method of matched asymptotic expansions is employed to approximate the periodic orbit of a relaxation oscillator. As an introduction, in chapter 2 the asymptotic analysis of Van der Pol's equation is carried out in all detail. The problem exhibits all features characteristic for a relaxation oscillation. From this case study one may learn how to handle other or more generally formulated relaxation oscillations. In the survey special attention is given to biological and chemical relaxation oscillators. In chapter 2 a general definition of a relaxation oscillation is formulated.

Keywords

Approximation bifurcation differential equation diffusion dynamical systems modeling

Authors and affiliations

  • Johan Grasman
    • 1
  1. 1.Department of MathematicsState University of UtrechtUtrechtThe Netherlands

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-1056-6
  • Copyright Information Springer-Verlag New York, Inc. 1987
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-96513-0
  • Online ISBN 978-1-4612-1056-6
  • Series Print ISSN 0066-5452
  • Buy this book on publisher's site