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Asymptotic Methods For Relaxation Oscillations And Applications
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Book Synopsis Asymptotic Methods for Relaxation Oscillations and Applications by : Johan Grasman
Download or read book Asymptotic Methods for Relaxation Oscillations and Applications written by Johan Grasman and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 229 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Asymptotic Methods in the Theory of Non-linear Oscillations by : Nikolaĭ Nikolaevich Bogoli︠u︡bov
Download or read book Asymptotic Methods in the Theory of Non-linear Oscillations written by Nikolaĭ Nikolaevich Bogoli︠u︡bov and published by CRC Press. This book was released on 1961 with total page 556 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Differential Equations with Small Parameters and Relaxation Oscillations by : E. Mishchenko
Download or read book Differential Equations with Small Parameters and Relaxation Oscillations written by E. Mishchenko and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 235 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Methods and Applications of Singular Perturbations by : Ferdinand Verhulst
Download or read book Methods and Applications of Singular Perturbations written by Ferdinand Verhulst and published by Springer Science & Business Media. This book was released on 2006-06-04 with total page 332 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains well-chosen examples and exercises A student-friendly introduction that follows a workbook type approach
Book Synopsis Asymptotic Methods in Singularly Perturbed Systems by : Evgeniĭ Frolovich Mishchenko
Download or read book Asymptotic Methods in Singularly Perturbed Systems written by Evgeniĭ Frolovich Mishchenko and published by . This book was released on 1994 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: Furthers the construction of a full asymptotic theory of relaxation oscillations begun by earlier authors, and contains the results of a number of new problems, especially in systems of parabolic partial differential equations. Considers a singularly perturbed system to be one in which as the parame
Book Synopsis Encyclopaedia of Mathematics (set) by : Michiel Hazewinkel
Download or read book Encyclopaedia of Mathematics (set) written by Michiel Hazewinkel and published by Springer Science & Business Media. This book was released on 1994-02-28 with total page 982 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. Jones Publisher :Springer Science & Business Media ISBN 13 :1461301173 Total Pages :278 pages Book Rating :4.4/5 (613 download)
Book Synopsis Multiple-Time-Scale Dynamical Systems by : Christopher K.R.T. Jones
Download or read book Multiple-Time-Scale Dynamical Systems written by Christopher K.R.T. Jones and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 278 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Encyclopaedia of Mathematics by : Michiel Hazewinkel
Download or read book Encyclopaedia of Mathematics written by Michiel Hazewinkel and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 499 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Boundary Function Method for Singular Perturbed Problems by : Adelaida B. Vasil'eva
Download or read book The Boundary Function Method for Singular Perturbed Problems written by Adelaida B. Vasil'eva and published by SIAM. This book was released on 1995-01-01 with total page 231 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted solely to the boundary function method, which is one of the asymptotic methods.
Book Synopsis Encyclopaedia of Mathematics by : M. Hazewinkel
Download or read book Encyclopaedia of Mathematics written by M. Hazewinkel and published by Springer. This book was released on 2013-12-01 with total page 967 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Relaxation Oscillations in Mathematical Models of Ecology by : A. I︠U︡ Kolesov
Download or read book Relaxation Oscillations in Mathematical Models of Ecology written by A. I︠U︡ Kolesov and published by American Mathematical Soc.. This book was released on 1995 with total page 140 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Differential Equations and Applications by :
Download or read book Differential Equations and Applications written by and published by Elsevier. This book was released on 1978-01-01 with total page 263 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential Equations and Applications
Book Synopsis Applied Asymptotic Methods in Nonlinear Oscillations by : Yuri A. Mitropolsky
Download or read book Applied Asymptotic Methods in Nonlinear Oscillations written by Yuri A. Mitropolsky and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 352 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many dynamical systems are described by differential equations that can be separated into one part, containing linear terms with constant coefficients, and a second part, relatively small compared with the first, containing nonlinear terms. Such a system is said to be weakly nonlinear. The small terms rendering the system nonlinear are referred to as perturbations. A weakly nonlinear system is called quasi-linear and is governed by quasi-linear differential equations. We will be interested in systems that reduce to harmonic oscillators in the absence of perturbations. This book is devoted primarily to applied asymptotic methods in nonlinear oscillations which are associated with the names of N. M. Krylov, N. N. Bogoli ubov and Yu. A. Mitropolskii. The advantages of the present methods are their simplicity, especially for computing higher approximations, and their applicability to a large class of quasi-linear problems. In this book, we confine ourselves basi cally to the scheme proposed by Krylov, Bogoliubov as stated in the monographs [6,211. We use these methods, and also develop and improve them for solving new problems and new classes of nonlinear differential equations. Although these methods have many applications in Mechanics, Physics and Technique, we will illustrate them only with examples which clearly show their strength and which are themselves of great interest. A certain amount of more advanced material has also been included, making the book suitable for a senior elective or a beginning graduate course on nonlinear oscillations.
Book Synopsis Multiple Scale and Singular Perturbation Methods by : J.K. Kevorkian
Download or read book Multiple Scale and Singular Perturbation Methods written by J.K. Kevorkian and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 642 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a revised and updated version, including a substantial portion of new material, of our text Perturbation Methods in Applied Mathematics (Springer Verlag, 1981). We present the material at a level that assumes some familiarity with the basics of ordinary and partial differential equations. Some of the more advanced ideas are reviewed as needed; therefore this book can serve as a text in either an advanced undergraduate course or a graduate-level course on the subject. Perturbation methods, first used by astronomers to predict the effects of small disturbances on the nominal motions of celestial bodies, have now become widely used analytical tools in virtually all branches of science. A problem lends itself to perturbation analysis if it is "close" to a simpler problem that can be solved exactly. Typically, this closeness is measured by the occurrence of a small dimensionless parameter, E, in the governing system (consisting of differential equations and boundary conditions) so that for E = 0 the resulting system is exactly solvable. The main mathematical tool used is asymptotic expansion with respect to a suitable asymptotic sequence of functions of E. In a regular perturbation problem, a straightforward procedure leads to a system of differential equations and boundary conditions for each term in the asymptotic expansion. This system can be solved recursively, and the accuracy of the result improves as E gets smaller, for all values of the independent variables throughout the domain of interest. We discuss regular perturbation problems in the first chapter.
Book Synopsis Spectral Methods for Incompressible Viscous Flow by : Roger Peyret
Download or read book Spectral Methods for Incompressible Viscous Flow written by Roger Peyret and published by Springer Science & Business Media. This book was released on 2002-03-28 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: This well-written book explains the theory of spectral methods and their application to the computation of viscous incompressible fluid flow, in clear and elementary terms. With many examples throughout, the work will be useful to those teaching at the graduate level, as well as to researchers working in the area.
Book Synopsis Variational Methods for Structural Optimization by : Andrej Cherkaev
Download or read book Variational Methods for Structural Optimization written by Andrej Cherkaev and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 561 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book bridges a gap between a rigorous mathematical approach to variational problems and the practical use of algorithms of structural optimization in engineering applications. The foundations of structural optimization are presented in sufficiently simple form as to make them available for practical use.
Book Synopsis Direct Methods in the Calculus of Variations by : Bernard Dacorogna
Download or read book Direct Methods in the Calculus of Variations written by Bernard Dacorogna and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: In recent years there has been a considerable renewal of interest in the clas sical problems of the calculus of variations, both from the point of view of mathematics and of applications. Some of the most powerful tools for proving existence of minima for such problems are known as direct methods. They are often the only available ones, particularly for vectorial problems. It is the aim of this book to present them. These methods were introduced by Tonelli, following earlier work of Hilbert and Lebesgue. Although there are excellent books on calculus of variations and on direct methods, there are recent important developments which cannot be found in these books; in particular, those dealing with vector valued functions and relaxation of non convex problems. These two last ones are important in appli cations to nonlinear elasticity, optimal design . . . . In these fields the variational methods are particularly effective. Part of the mathematical developments and of the renewal of interest in these methods finds its motivations in nonlinear elasticity. Moreover, one of the recent important contributions to nonlinear analysis has been the study of the behaviour of nonlinear functionals un der various types of convergence, particularly the weak convergence. Two well studied theories have now been developed, namely f-convergence and compen sated compactness. They both include as a particular case the direct methods of the calculus of variations, but they are also, both, inspired and have as main examples these direct methods.
