Asymptotic Methods for Relaxation Oscillations and Applications

1. Front Matter

   Pages N2-xiii

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2. Introduction

   • Johan Grasman

   Pages 1-22

3. Free Oscillation

   • Johan Grasman

   Pages 23-114

4. Forced Oscillation and Mutual Entrainment

   • Johan Grasman

   Pages 115-150

5. The Van der Pol Oscillator with a Sinusoidal Forcing Term

   • Johan Grasman

   Pages 151-199

6. Back Matter

   Pages 201-223

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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.

• Approximation
• bifurcation
• differential equation
• diffusion
• dynamical systems
• modeling

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

  Johan Grasman

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