Asymptotic Methods for Relaxation Oscillations and Applications

Softcover Nachdruck of the original 1. Auflage 1987
ISBN: 978-0-387-96513-0
Verlag: Springer

Buch, Englisch, Band 63, 227 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 371 g

Reihe: Applied Mathematical Sciences

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.

Zielgruppe

Research

Autoren/Hrsg.

Grasman, Johan

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Computeranwendungen in der Mathematik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

1. Introduction.- 1.1 The Van der Pol oscillator.- 1.2 Mechanical prototypes of relaxation oscillators.- 1.3 Relaxation oscillations in physics and biology.- 1.4 Discontinuous approximations.- 1.5 Matched asymptotic expansions.- 1.6 Forced oscillations.- 1.7 Mutual entrainment.- 2 Free oscillation.- 2.1 Autonomous relaxation oscillation: definition and existence.- 2.2 Asymptotic solution of the Van der Pol equation.- 2.3 The Volterra-Lotka equations.- 2.4 Chemical oscillations.- 2.5 Bifurcation of the Van der Pol equation with a constant forcing term.- 2.6 Stochastic and chaotic oscillations.- 3. Forced oscillation and mutual entrainment.- 3.1 Modeling coupled oscillations.- 3.2 A rigorous theory for weakly coupled oscillators.- 3.3 Coupling of two oscillators.- 4. The Van der Pol oscillator with a sinusoidal forcing term.- 4.1 Qualitative methods of analysis.- 4.2 Asymptotic solution of the Van der Pol equation with a moderate forcing term.- 4.2 Asymptotic solution of the Van der Pol equation with a large forcing term.- 4.3 Asymptotic solution of the Van der Pol equation with a large forcing term.- Appendices.- A: Asymptotics of some special functions.- B: Asymptotic ordering and expansions.- C: Concepts of the theory of dynamical systems.- D: Stochastic differential equations and diffusion approximations.- Literature.- Author Index.