Averaging Methods in Nonlinear Dynamical Systems

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.

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)