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Asymptotic Methods For Relaxation Oscillations
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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 Asymptotic Methods for relaxation oscillations by : Johan Grasman
Download or read book Asymptotic Methods for relaxation oscillations written by Johan Grasman and published by . This book was released on 1977 with total page 19 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Asymptotic Representation of Relaxation Oscillations in Lasers by : Elena V. Grigorieva
Download or read book Asymptotic Representation of Relaxation Oscillations in Lasers written by Elena V. Grigorieva and published by Birkhäuser. This book was released on 2016-11-09 with total page 233 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
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 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 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 Asymptotic Methods in Singularly Perturbed Systems by : E.F. Mishchenko
Download or read book Asymptotic Methods in Singularly Perturbed Systems written by E.F. Mishchenko and published by Springer. This book was released on 1994-09-14 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: There are many books devoted to ordinary differential equations con taining small parameters (small perturbations). The investigation of the dependence of solutions, in a finite time interval, on regular perturbations (the small parameter regularly appears on the right-hand sides of the equa tions) was carried out by Poincare and was practically completed long ago. However, problems connected with singular perturbations still attract the attention of mathematicians. This is what we understand by a singularly perturbed system: a system of differential equations dependent on a small parameter is said to be singularly perturbed if, as the parameter tends to zero, Cauchy's resolvent operator for the main range of time values and initial conditions from bounded sets (or the Poincare operator) converges, in a suitable topology, to a limit object acting in a space of smaller dimension. In different cases this general idea of a singularly perturbed system becomes specific and leads to numerous important and interesting problems. A certain class of these problems was only recently considered in mono graphic literature. This class includes problems connected with the so-called relaxation oscillations, a phenomenon well known to physicists, mechani cians, chemists, and ecologists. Van der Pol, Andronov, Haag, Dorodnitsyn, Stoker, Zheleztsov and others were the first to study relaxation oscillations. A comprehensive study of this phenomenon is hindered by considerable mathematical difficulties and requires the development of new asymptotic methods in the theory of differential equations. These methods, interesting in themselves, lead to the statement of new mathematical problems.
Book Synopsis Asymptotic Methods in the Theory of Non-linear Oscillations by : N. N. Bogoliubov
Download or read book Asymptotic Methods in the Theory of Non-linear Oscillations written by N. N. Bogoliubov and published by . This book was released on 1961 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Introduction to Nonlinear Oscillations by : Vladimir I. Nekorkin
Download or read book Introduction to Nonlinear Oscillations written by Vladimir I. Nekorkin and published by John Wiley & Sons. This book was released on 2015-04-01 with total page 264 pages. Available in PDF, EPUB and Kindle. Book excerpt: A systematic outline of the basic theory of oscillations, combining several tools in a single textbook. The author explains fundamental ideas and methods, while equally aiming to teach students the techniques of solving specific (practical) or more complex problems. Following an introduction to fundamental notions and concepts of modern nonlinear dynamics, the text goes on to set out the basics of stability theory, as well as bifurcation theory in one and two-dimensional cases. Foundations of asymptotic methods and the theory of relaxation oscillations are presented, with much attention paid to a method of mappings and its applications. With each chapter including exercises and solutions, including computer problems, this book can be used in courses on oscillation theory for physics and engineering students. It also serves as a good reference for students and scientists in computational neuroscience.
Book Synopsis Asymptotic Solution of Relaxation Oscillation Problems by : Edgar N. Gilbert
Download or read book Asymptotic Solution of Relaxation Oscillation Problems written by Edgar N. Gilbert and published by . This book was released on 1948 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Asymptotic Methods in the Theory of Nonlinear Oscillations by : Nikolaĭ Nikolaevich Bogoli︠u︡bov
Download or read book Asymptotic Methods in the Theory of Nonlinear Oscillations written by Nikolaĭ Nikolaevich Bogoli︠u︡bov and published by . This book was released on 1955 with total page 898 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Asymptotic methods in the theory of non-linear oscillations by : Nikolaj N. Bogoljubov
Download or read book Asymptotic methods in the theory of non-linear oscillations written by Nikolaj N. Bogoljubov and published by . This book was released on 1985 with total page 537 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Applied Asymptotic Methods in Nonlinear Oscillations by : I︠U︡riĭ Alekseevich Mitropolʹskiĭ
Download or read book Applied Asymptotic Methods in Nonlinear Oscillations written by I︠U︡riĭ Alekseevich Mitropolʹskiĭ and published by . This book was released on 1994 with total page 411 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Asymptotic Methods in the Theory of Non-linear Oscillations by :
Download or read book Asymptotic Methods in the Theory of Non-linear Oscillations written by and published by . This book was released on 1961 with total page 537 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Mathematics of Complexity and Dynamical Systems by : Robert A. Meyers
Download or read book Mathematics of Complexity and Dynamical Systems written by Robert A. Meyers and published by Springer Science & Business Media. This book was released on 2011-10-05 with total page 1885 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Complexity and Dynamical Systems is an authoritative reference to the basic tools and concepts of complexity, systems theory, and dynamical systems from the perspective of pure and applied mathematics. Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through self-organization, e.g. the spontaneous formation of temporal, spatial or functional structures. These systems are often characterized by extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even completely deterministic. The more than 100 entries in this wide-ranging, single source work provide a comprehensive explication of the theory and applications of mathematical complexity, covering ergodic theory, fractals and multifractals, dynamical systems, perturbation theory, solitons, systems and control theory, and related topics. Mathematics of Complexity and Dynamical Systems is an essential reference for all those interested in mathematical complexity, from undergraduate and graduate students up through professional researchers.
Book Synopsis Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications by : Johan Grasman
Download or read book Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications written by Johan Grasman and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 224 pages. Available in PDF, EPUB and Kindle. Book excerpt: Asymptotic methods are of great importance for practical applications, especially in dealing with boundary value problems for small stochastic perturbations. This book deals with nonlinear dynamical systems perturbed by noise. It addresses problems in which noise leads to qualitative changes, escape from the attraction domain, or extinction in population dynamics. The most likely exit point and expected escape time are determined with singular perturbation methods for the corresponding Fokker-Planck equation. The authors indicate how their techniques relate to the Itô calculus applied to the Langevin equation. The book will be useful to researchers and graduate students.
