Normal Forms and Unfoldings for Local Dynamical Systems / Edition 1

Normal Forms and Unfoldings for Local Dynamical Systems / Edition 1

by James Murdock
     
 

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… See more details below

Overview

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 The book contains previously unpublished results in both areas (algebraic and geometrical) and includes suggestions for further research.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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Product Details

ISBN-13:
9781441930132
Publisher:
Springer New York
Publication date:
12/01/2010
Series:
Springer Monographs in Mathematics Series
Edition description:
Softcover reprint of hardcover 1st ed. 2003
Pages:
500
Product dimensions:
1.04(w) x 9.21(h) x 6.14(d)

Meet the Author

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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