Synopses & Reviews
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 The book contains previously unpublished results in both areas (algebraic and geometrical) and includes suggestions for further research.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.
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)
Includes bibliographical references (p. -487) and index.
The subject of local dynamical systems is concerned with the following two questions: 1. Given an nxn matrix A, describe the behavior, in a neighborhood of the origin, of the solutions of all systems of di?erential equations having a rest point at the origin with linear part Ax, that is, all systems of the form x ? = Ax+..., n where x? R and the dots denote terms of quadratic and higher order. 2. Describethebehavior(neartheorigin)ofallsystemsclosetoasystem of the type just described. To answer these questions, the following steps are employed: 1. A normal form is obtained for the general system with linear part Ax. The normal form is intended to be the simplest form into which any system of the intended type can be transformed by changing the coordinates in a prescribed manner. 2. An unfolding of the normal form is obtained. This is intended to be the simplest form into which all systems close to the original s- tem can be transformed. It will contain parameters, called unfolding parameters, that are not present in the normal form found in step 1. vi Preface 3. The normal form, or its unfolding, is truncated at some degree k, and the behavior of the truncated system is studied."
The largest part of this book is devoted to normal forms, divided into semisimple theory, applied when the linear part is diagonalizable, and the general theory, applied when the linear part is the sum of the semisimple and nilpotent matrices. One of the objectives of this book is to develop all of the necessary theory 'from scratch' in just the form that is needed for the application to normal forms, with as little unnecessary terminology as possible. The intended audience is Ph.D. students and researchers in applied mathematics, theoretical physics, and advanced engineering, though in principle it could be read by anyone with a sufficient background in linear algebra and differential equations.
This is the most thorough treatment of normal forms currently existing in book form. There is a substantial gap between elementary treatments in textbooks and advanced research papers on normal forms. This book develops all the necessary theory 'from scratch' in just the form that is needed for the application to normal forms, with as little unnecessary terminology as possible.
Table of Contents
Preface.- 1. Two Examples.- 2. The splitting problem for linear operators.- 3. Linear Normal Forms.- 4. Nonlinear Normal Forms.- 5. Geometrical Structures in Normal Forms.- 6. Selected Topics in Local Bifurcation Theory.- Appendix A: Rings.- Appendix B: Modules.- Appendix C: Format 2b: Generated Recursive (Hori).- Appendix D: Format 2c: Generated Recursive (Deprit).- Appendix E: On Some Algorithms in Linear Algebra.- Bibliography.- Index.