Averaging Methods in Nonlinear Dynamical Systems

ISBN-13: 9780387489162 / Angielski / Twarda / 2007 / 434 str.

Jan A. Sanders; Ferdinand Verhulst; James Murdock

Opis

Perturbation theory and in particular normal form theory has shown strong growth in recent decades. This book is a drastic revision of the first edition of the averaging book. The updated chapters represent new insights in averaging, in particular its relation with dynamical systems and the theory of normal forms. Also new are survey appendices on invariant manifolds. One of the most striking features of the book is the collection of examples, which range from the very simple to some that are elaborate, realistic, and of considerable practical importance. Most of them are presented in careful detail and are illustrated with illuminating diagrams.

Recenzje

From the reviews of the second edition:

"This monograph is a second edition comprising a thorough revision and an expansion of the first one ... . Thus, the reader is exposed to the practice of examining both concrete applications ... and theory. ... comprehensive content is extremely well written. ... The presentation is self-contained, thus offering a high level and convenient exposition to those who wish to study the subject matter as a whole. ... The text introduces particular notations which are not too common in the literature." (Zvi Artstein, Mathematical Reviews, Issue 2008 h)

"The new book will be an ideal place to learn about averaging, including what's new in the last quarter century ... . Overall, the authors are to be commended for writing this timely and important piece of scholarship, which should teach many about their topic and related analysis." (Robert E. O' Malley, Jr., Siam Review, Vol. 51 (1), 2009)

"This monograph is a revised and expanded second edition of the original text from 1985 by the first two authors. This first edition has since then become one of the standard references for the modern theory of averaging and singular perturbations of ordinary differential equations. Without doubt the second edition will continue this legacy. ... The book is well-written with full proves and a wealth of enlightening examples. It can only be warmly recommended to everybody working in this field ... ." (G. Teschl, Monatshefte für Mathematik, Vol. 156 (4), April, 2009)

Spis treści

Basic Material and Asymptotics.- Averaging: the Periodic Case.- Methodology of Averaging.- Averaging: the General Case.- Attraction.- Periodic Averaging and Hyperbolicity.- Averaging over Angles.- Passage Through Resonance.- From Averaging to Normal Forms.- Hamiltonian Normal Form Theory.- Classical (First-Level) Normal Form Theory.- Nilpotent (Classical) Normal Form.- Higher-Level Normal Form Theory.- The History of the Theory of Averaging.- A 4-Dimensional Example of Hopf Bifurcation.- Invariant Manifolds by Averaging.- Some Elementary Exercises in Celestial Mechanics.- On Averaging Methods for Partial Differential Equations.

Blurb

Perturbation theory and in particular normal form theory has shown strong growth during the last decades. So it is not surprising that the authors have presented an extensive revision of the first edition of the Averaging Methods in Nonlinear Dynamical Systems book. There are many changes, corrections and updates in chapters on Basic Material and Asymptotics, Averaging, and Attraction. Chapters on Periodic Averaging and Hyperbolicity, Classical (first level) Normal Form Theory, Nilpotent (classical) Normal Form, and Higher Level Normal Form Theory are entirely new and represent new insights in averaging, in particular its relation with dynamical systems and the theory of normal forms. Also new are surveys on invariant manifolds in Appendix C and averaging for PDEs in Appendix E. Since the first edition, the book has expanded in length and the third author, James Murdock has been added.

Review of First Edition

"One of the most striking features of the book is the nice collection of examples, which range from the very simple to some that are elaborate, realistic, and of considerable practical importance. Most of them are presented in careful detail and are illustrated with profuse, illuminating diagrams." - Mathematical Reviews