##
**Asymptotic approaches in nonlinear dynamics. New trends and applications.**
*(English)*
Zbl 0910.70001

Springer Series in Synergetics. Berlin: Springer. xi, 310 p. (1998).

This book gives an introduction into the field of asymptotic simplification of problems of the theory of oscillations. On the other hand there is no lack in good books treating the application of asymptotic methods to problems of nonlinear mechanics. Hence quite naturally the question arises: what specific aspects justify the publication of a new book?

A positive answer in the present case is quite obvious because the great asset of this book is that two of the authors are from countries (Russia, Ukraine) where the asymptotic theories for the treatment of nonlinear oscillations have been mostly developed during the past century and are still treated nowadays on a very high level. However, a great part of the corresponding literature is still published in the native language of these countries and hence not easily accessible. In this book, many results are reported in English language for the first time in a unified and pedagogically well-done way.

The content of the book covers the basic principles of asymptotics and deals on the one hand with traditional approaches, such as regular and singular perturbations, averaging and homogenization. On the other hand, the authors present some less well-known approaches like Padé approximants, the distributional approach and the method of boundary perturbations. Pedagogically, it is very appealing that almost all concepts are introduced and explained by means of well selected examples from nonlinear mechanics which are easy to understand and which partly are worked out in a very detailed manner.

This book can be highly recommended to every scientist working in the field of asymptotic methods in nonlinear dynamics.

A positive answer in the present case is quite obvious because the great asset of this book is that two of the authors are from countries (Russia, Ukraine) where the asymptotic theories for the treatment of nonlinear oscillations have been mostly developed during the past century and are still treated nowadays on a very high level. However, a great part of the corresponding literature is still published in the native language of these countries and hence not easily accessible. In this book, many results are reported in English language for the first time in a unified and pedagogically well-done way.

The content of the book covers the basic principles of asymptotics and deals on the one hand with traditional approaches, such as regular and singular perturbations, averaging and homogenization. On the other hand, the authors present some less well-known approaches like Padé approximants, the distributional approach and the method of boundary perturbations. Pedagogically, it is very appealing that almost all concepts are introduced and explained by means of well selected examples from nonlinear mechanics which are easy to understand and which partly are worked out in a very detailed manner.

This book can be highly recommended to every scientist working in the field of asymptotic methods in nonlinear dynamics.

Reviewer: H.Troger (Wien)

### MSC:

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |

70K99 | Nonlinear dynamics in mechanics |

34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |

34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |