2.1 AUTONOMOUS RELAXATION OSCILLATION: DEFINITION AND EXISTENCE In this section we will formulate a definition of relaxation oscillations. In the introduction a notion of relaxation oscillations evolved, which is more or less a ...

Author: Johan Grasman

Publisher: Springer Science & Business Media

ISBN: 9781461210566

Category: Science

Page: 227

View: 617

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.

I have stressed that the proposed workbook format is very suitable for singular perturbation problems, but I hope that the ... Also there are books available on this topic, such as Asymptotic Methods for Relaxation Oscillations and ...

Author: Ferdinand Verhulst

Publisher: Springer Science & Business Media

ISBN: 9780387283135

Category: Mathematics

Page: 328

View: 182

Contains well-chosen examples and exercises A student-friendly introduction that follows a workbook type approach

For weakly - forced and weaklycoupled relaxation oscillations one can derive formulas for synchronization and phase shift ... References ( A1 ) [ A1 ] GRASMAN , J .: Asymptotic methods for relaxation oscillations and applications ...

Author: Michiel Hazewinkel

Publisher: Springer Science & Business Media

ISBN: 1556080107

Category: Mathematics

Page: 982

View: 691

The Encyclopaedia of Mathematics is the most up-to-date, authoritative and comprehensive English-language work of reference in mathematics which exists today. With over 7,000 articles from `A-integral' to `Zygmund Class of Functions', supplemented with a wealth of complementary information, and an index volume providing thorough cross-referencing of entries of related interest, the Encyclopaedia of Mathematics offers an immediate source of reference to mathematical definitions, concepts, explanations, surveys, examples, terminology and methods. The depth and breadth of content and the straightforward, careful presentation of the information, with the emphasis on accessibility, makes the Encyclopaedia of Mathematics an immensely useful tool for all mathematicians and other scientists who use, or are confronted by, mathematics in their work. The Enclyclopaedia of Mathematics provides, without doubt, a reference source of mathematical knowledge which is unsurpassed in value and usefulness. It can be highly recommended for use in libraries of universities, research institutes, colleges and even schools.

Author: Christopher K.R.T. JonesPublish On: 2012-12-06

[9] N. FENICHEL, Geometric singular perturbation theory, J. Diff. Eq. 31, pp. 53-98 (1979). [10] J. GRASMAN, Asymptotic methods for relaxation oscillations and applications, Springer, New York (1987). [11] J. GUCKENHEIMER AND P. HoLMES, ...

Author: Christopher K.R.T. Jones

Publisher: Springer Science & Business Media

ISBN: 9781461301172

Category: Mathematics

Page: 273

View: 207

Systems with sub-processes evolving on many different time scales are ubiquitous in applications: chemical reactions, electro-optical and neuro-biological systems, to name just a few. This volume contains papers that expose the state of the art in mathematical techniques for analyzing such systems. Recently developed geometric ideas are highlighted in this work that includes a theory of relaxation-oscillation phenomena in higher dimensional phase spaces. Subtle exponentially small effects result from singular perturbations implicit in certain multiple time scale systems. Their role in the slow motion of fronts, bifurcations, and jumping between invariant tori are all explored here. Neurobiology has played a particularly stimulating role in the development of these techniques and one paper is directed specifically at applying geometric singular perturbation theory to reveal the synchrony in networks of neural oscillators.

important phenomena are determined by asymptotics lurking beyond all algebraic orders O ( EN ) . ... J. Grasman , Asymptotic Methods for Relaxation Oscillations and Applications , SpringerVerlag , New York , 1987 . 15.

Author: Jane Cronin

Publisher: American Mathematical Soc.

ISBN: 082186761X

Category: Mathematics

Page: 204

View: 621

To understand multiscale phenomena, it is essential to employ asymptotic methods to construct approximate solutions and to design effective computational algorithms. This volume consists of articles based on the AMS Short Course in Singular Perturbations held at the annual Joint Mathematics Meetings in Baltimore (MD). Leading experts discussed the following topics which they expand upon in the book: boundary layer theory, matched expansions, multiple scales, geometric theory, computational techniques, and applications in physiology and dynamic metastability. Readers will find that this text offers an up-to-date survey of this important field with numerous references to the current literature, both pure and applied.

Author: Adelaida B. Vasil'evaPublish On: 1995-01-01

1092-1101. , Asymptotic solution of a quasistatic thermoelasticity problem for a slender rod, J. Appl. Math. ... [56] J. GRASMAN, Asymptotic Methods for Relaxation Oscillations and Applications, Springer-Verlag, Berlin, Heidelberg, ...

Author: Adelaida B. Vasil'eva

Publisher: SIAM

ISBN: 9780898713336

Category: Mathematics

Page: 231

View: 347

This book is devoted solely to the boundary function method, which is one of the asymptotic methods.

An asymptotic expansion for the periodic solution of the Van der Pol equation with v >> 1 has been given by DORODNICYN [6]. Because of the changing behaviour of relaxations oscillations one has to apply methods different from those for ...

Studies on Divergent Series and Summability and the Asymptotic Developments of Functions Defined by Maclaurin ... “On the method of matched asymptotic expansions. ... Asymptotic Methods for Relaxation Oscillations and Applications.

Author: Mark H. Holmes

Publisher: Springer Science & Business Media

ISBN: 9781461253471

Category: Mathematics

Page: 356

View: 559

This introductory graduate text is based on a graduate course the author has taught repeatedly over the last ten years to students in applied mathematics, engineering sciences, and physics. Each chapter begins with an introductory development involving ordinary differential equations, and goes on to cover such traditional topics as boundary layers and multiple scales. However, it also contains material arising from current research interest, including homogenisation, slender body theory, symbolic computing, and discrete equations. Many of the excellent exercises are derived from problems of up-to-date research and are drawn from a wide range of application areas.