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Normal Forms and Unfoldings for Local Dynamical Systems (Springer Monographs in Mathematics)by James Murdock
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 The book contains previously unpublished results in both areas (algebraic and geometrical) and includes suggestions for further research.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.
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.
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.
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.
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