Normal Forms and Unfoldings for Local Dynamical Systems
The subject of local dynamical systems is concerned with the following two questions: 1. Given an n×n matrix A, describe the behavior, in a neighborhood of the origin, of the solutions of all systems of differential 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 system 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.
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Normal Forms and Unfoldings for Local Dynamical Systems
The subject of local dynamical systems is concerned with the following two questions: 1. Given an n×n matrix A, describe the behavior, in a neighborhood of the origin, of the solutions of all systems of differential 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 system 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.
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Normal Forms and Unfoldings for Local Dynamical Systems

Normal Forms and Unfoldings for Local Dynamical Systems

by James Murdock
Normal Forms and Unfoldings for Local Dynamical Systems

Normal Forms and Unfoldings for Local Dynamical Systems

by James Murdock

Hardcover(2003)

$54.99 
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Overview

The subject of local dynamical systems is concerned with the following two questions: 1. Given an n×n matrix A, describe the behavior, in a neighborhood of the origin, of the solutions of all systems of differential 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 system 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.

Product Details

ISBN-13: 9780387954646
Publisher: Springer New York
Publication date: 11/20/2002
Series: Springer Monographs in Mathematics
Edition description: 2003
Pages: 500
Product dimensions: 6.10(w) x 9.25(h) x 0.04(d)

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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