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Differential Equations with Small Parameters and Relaxation Oscillations

Author : E. Mishchenko

Publisher : Springer Science & Business Media

Page : 235 pages

File Size : 31,14 MB

Release : 2013-03-09

Category : Science

ISBN : 1461590477

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Book Description

A large amount of work has been done on ordinary differ ential equations with small parameters multiplying deriv atives. This book investigates questions related to the asymptotic calculation of relaxation oscillations, which are periodic solutions formed of sections of both sl- and fast-motion parts of phase trajectories. A detailed discussion of solutions of differential equations involving small parameters is given for regions near singular points. The main results examined were obtained by L.S. Pontryagin and the authors. Other works have also been taken into account: A.A. Dorodnitsyn's investigations of Van der Pol's equation, results obtained by N.A. Zheleztsov and L.V. Rodygin concerning relaxation oscillations in electronic devices, and results due to A.N. Tikhonov and A.B. Vasil'eva concerning differential equations with small parameters multiplying certain derivatives. E.F. Mishchenko N. Kh. Rozov v CONTENTS Chapter I. Dependence of Solutions on Small Parameters. Applications of Relaxation Oscillations 1. Smooth Dependence. Poincare's Theorem . 1 2. Dependence of Solutions on a Parameter, on an Infinite Time Interval 3 3. Equations with Small Parameters 4 Multiplying Derivatives 4. Second-Order Systems. Fast and Slow Motion.







Asymptotic Representation of Relaxation Oscillations in Lasers

Author : Elena V. Grigorieva

Publisher : Birkhäuser

Page : 233 pages

File Size : 13,75 MB

Release : 2016-11-09

Category : Science

ISBN : 3319428608

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Book Description

In this book we analyze relaxation oscillations in models of lasers with nonlinear elements controlling light dynamics. The models are based on rate equations taking into account periodic modulation of parameters, optoelectronic delayed feedback, mutual coupling between lasers, intermodal interaction and other factors. With the aim to study relaxation oscillations we present the special asymptotic method of integration for ordinary differential equations and differential-difference equations. As a result, they are reduced to discrete maps. Analyzing the maps we describe analytically such nonlinear phenomena in lasers as multistability of large-amplitude relaxation cycles, bifurcations of cycles, controlled switching of regimes, phase synchronization in an ensemble of coupled systems and others. The book can be fruitful for students and technicians in nonlinear laser dynamics and in differential equations.



Relaxation Oscillations in Mathematical Models of Ecology

Author : A. I︠U︡ Kolesov

Publisher : American Mathematical Soc.

Page : 140 pages

File Size : 30,72 MB

Release : 1995

Category : Mathematics

ISBN : 9780821804100

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Book Description

This book presents for the first time a systematic exposition of techniques for constructing relaxation oscillations and methods for investigating stability properties of certain classes of systems with delay. The authors bring out some of the distinctive features that have no analogues in relaxation systems of ordinary differential equations. The exposition provides analysis of significant examples from biophysics, mathematical ecology, and quantum physics that elucidate important patterns. Many unsolved problems are posed. The book would appeal to researchers and specialists interested in the theory and applications of relaxation oscillations.



Dynamical Systems V

Author : V.I. Arnold

Publisher : Springer Science & Business Media

Page : 279 pages

File Size : 43,54 MB

Release : 2013-12-01

Category : Mathematics

ISBN : 3642578845

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Book Description

Bifurcation theory and catastrophe theory are two well-known areas within the field of dynamical systems. Both are studies of smooth systems, focusing on properties that seem to be manifestly non-smooth. Bifurcation theory is concerned with the sudden changes that occur in a system when one or more parameters are varied. Examples of such are familiar to students of differential equations, from phase portraits. Understanding the bifurcations of the differential equations that describe real physical systems provides important information about the behavior of the systems. Catastrophe theory became quite famous during the 1970's, mostly because of the sensation caused by the usually less than rigorous applications of its principal ideas to "hot topics", such as the characterization of personalities and the difference between a "genius" and a "maniac". Catastrophe theory is accurately described as singularity theory and its (genuine) applications. The authors of this book, previously published as Volume 5 of the Encyclopaedia, have given a masterly exposition of these two theories, with penetrating insight.





Multiple-Time-Scale Dynamical Systems

Author : Christopher K.R.T. Jones

Publisher : Springer Science & Business Media

Page : 278 pages

File Size : 23,60 MB

Release : 2012-12-06

Category : Mathematics

ISBN : 1461301173

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Book Description

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.



Using the Mathematics Literature

Author : Kristine K. Fowler

Publisher : CRC Press

Page : 412 pages

File Size : 38,59 MB

Release : 2004-05-25

Category : Language Arts & Disciplines

ISBN : 9780824750350

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Book Description

This reference serves as a reader-friendly guide to every basic tool and skill required in the mathematical library and helps mathematicians find resources in any format in the mathematics literature. It lists a wide range of standard texts, journals, review articles, newsgroups, and Internet and database tools for every major subfield in mathematics and details methods of access to primary literature sources of new research, applications, results, and techniques. Using the Mathematics Literature is the most comprehensive and up-to-date resource on mathematics literature in both print and electronic formats, presenting time-saving strategies for retrieval of the latest information.



Mathematical Aspects of Hodgkin-Huxley Neural Theory

Author : Jane Cronin

Publisher : Cambridge University Press

Page : 294 pages

File Size : 26,23 MB

Release : 1987-08-28

Category : Mathematics

ISBN : 9780521334822

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Book Description

This book is an introduction to the study of mathematical models of electrically active cells, which play an essential role in, for example, nerve conduction and cardiac functions. In the book, Dr Cronin synthesizes and reviews this material and provides a detailed discussion of the Hodgkin-Huxley model for nerve conduction, which forms the cornerstone of this body of work.