Normal Forms and Unfoldings for Local Dynamical Systems / Edition 1

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This book is about normal forms—the simplest form into which a dynamical system can be put for the purpose of studying its behavior in the neighborhood of a rest point—and about unfoldings—used to study the local bifurcations that the system can exhibit under perturbation. The book presents the advanced theory of normal forms, showing their interaction with representation theory, invariant theory, Groebner basis theory, and structure theory of rings and modules. A complete treatment is given both for the popular "inner product style" of normal forms and the less well known "sl(2) style" due to Cushman and Sanders, as well as the author's own "simplified" style. In addition, this book includes algorithms suitable for use with computer algebra systems for computing normal forms. The interaction between the algebraic structure of normal forms and their geometrical consequences is emphasized. The book contains previously unpublished results in both areas (algebraic and geometrical) and includes suggestions for further research.

The book begins with two nonlinear examples—one semisimple, one nilpotent—for which normal forms and unfoldings are computed by a variety of elementary methods. After treating some required topics in linear algebra, more advanced normal form methods are introduced, first in the context of linear normal forms for matrix perturbation theory, and then for nonlinear dynamical systems. Then the emphasis shifts to applications: geometric structures in normal forms, computation of unfoldings, and related topics in bifurcation theory.

This book will be useful to researchers and advanced students in dynamical systems, theoretical physics, and engineering.

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Editorial Reviews

From the Publisher

From the reviews:

"In the analysis of local dynamical systems … normal form theory plays an essential role. … this is a serious introduction to methods that have been developed in the last few decades. … This is a book that can be enjoyed on many levels, which is bound to give the reader new insights into the theory of normal forms and its applications." (Jan A. Sanders, Mathematical Reviews, Issue 2003 k)

"The book … aims to introduce both the algebraic structure of the coordinate transformations that are used in the normalization and … the geometric structure of the vector fields that are thus obtained. … The discussion … is the most lucid I have found to date. … The reader who expects to learn the basic ideas and techniques of normal form theory will find this book rewarding. Its algebraic approach is well suited to readers interested in automated computations of normal forms." (Kresimir Josic, Siam Review, Vol. 46 (4), 2004)

"Normal-form theory has become a celebrated topic which is widely used in nonlinear science. … This book certainly represents a very thorough treatment of the anatomy of normal-form transformations … . It may serve well as a reference work … and indeed the author achieves his stated aim of providing an encyclopedia of results and explanations which are not easily found in the existing literature." (Mark Groves, UK Nonlinear News, Issue 35, February, 2004)

"The theory of local dynamical systems studies neighbourhoods of a given equilibrium point, in particular the dynamical behaviour that is generically possible. … To my knowledge the monograph under review is the first successful attempt to deal with the ‘Elphick-Iooss’ inner product style and the ‘Cushman-Sanders’ sl(2) style at a larger scale. … the text successfully addresses computer-algebraic aspects of certain normal form computations that are useful for applications in concrete examples." (Henk Broer, Siam, November, 2003)

"This book is a treatise on normal forms and unfoldings of a dynamical system near a singular point. The goal is to lay down basic principles and this is … the main originality of this work. Moreover it is selfcontained. … The volume contains new results including some of the author. … This conceptually attractive and clearly written book is recommended." (A. Akutowicz, Zentralblatt MATH, Vol. 1014, 2003)

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Product Details

  • ISBN-13: 9780387954646
  • Publisher: Springer New York
  • Publication date: 11/20/2002
  • Series: Springer Monographs in Mathematics Series
  • Edition description: 2003
  • Edition number: 1
  • Pages: 500
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.13 (d)

Table of Contents

Two Examples * The Splitting Problem for Linear Operators * Linear Normal Forms * Nonlinear Normal Forms * Geometrical Structures in Normal Forms * Selected Topics in Local Bifurcation Theory

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