Normal Forms and Unfoldings for Local Dynamical Systems / Edition 1by James Murdock
Pub. Date: 11/20/2002
Publisher: Springer New York
This book is about normal formsthe simplest form into which a dynamical system can be put for the purpose of studying its behavior in the neighborhood of a rest pointand about unfoldingsused 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 The book contains previously unpublished results in both areas (algebraic and geometrical) and includes suggestions for further research.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 examplesone semisimple, one nilpotentfor 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.
Table of ContentsTwo 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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