Averaging methods in nonlinear dynamical systems. 2nd ed.

*(English)*Zbl 1128.34001
Applied Mathematical Sciences 59. New York, NY: Springer (ISBN 978-0-387-48916-2/hbk; 978-0-387-48918-6/ebook). xxi, 431 p. (2007).

This book is not only a revision of the first edition published 22 years ago (see Zbl 0586.34040), its extended content reflects also special developments in the theory of dynamical systems in the mentioned period: normal forms, invariant manifolds, perturbation theory.

The new material is organized in four chapters and two appendices. Chapter 6 is devoted to periodic averaging and hyperbolicity (interaction of the Morse-Smale theory including shadowing with averaging), Chapter 11 is entitled “classical (first level) normal form theory” and gives an introduction into the abstract formulation of the NFT, Chapter 12 treats nilpotent (classical) normal forms including computational aspects. Chapter 13 is concerned with higher-level NFT (continuation of the abstract treatment).

The new appendix C “invariant manifolds by averaging” considers the deformation of normally hyperbolic manifolds and different scenarios for the emergence of tori in some examples. Appendix E is devoted to averaging methods for partial differential equations. Since the results in that field are still fragmented, this section provides a survey on literature for weakly nonlinear PDE’s.

Last but not least it should be mentioned that James Murdock is included as an author.

The new material is organized in four chapters and two appendices. Chapter 6 is devoted to periodic averaging and hyperbolicity (interaction of the Morse-Smale theory including shadowing with averaging), Chapter 11 is entitled “classical (first level) normal form theory” and gives an introduction into the abstract formulation of the NFT, Chapter 12 treats nilpotent (classical) normal forms including computational aspects. Chapter 13 is concerned with higher-level NFT (continuation of the abstract treatment).

The new appendix C “invariant manifolds by averaging” considers the deformation of normally hyperbolic manifolds and different scenarios for the emergence of tori in some examples. Appendix E is devoted to averaging methods for partial differential equations. Since the results in that field are still fragmented, this section provides a survey on literature for weakly nonlinear PDE’s.

Last but not least it should be mentioned that James Murdock is included as an author.

Reviewer: Klaus R. Schneider (Berlin)

##### MSC:

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

37-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory |

34C29 | Averaging method for ordinary differential equations |

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

37Dxx | Dynamical systems with hyperbolic behavior |

37Gxx | Local and nonlocal bifurcation theory for dynamical systems |