Introduction

This paper is an introduction to perturbation analysis of differential equations and other nonlinear systems using normal form theory. Since it is an introduction a large part of the material presented will be classical, but the last section does contain some new applications. I hope that by understanding the classical results presented here the reader will be able to gain an entry into this rapidly evolving field. Since I am interested in applications of the theory to specific examples I will emphasize the development of the main computer algorithm and use of algebraic processors. This computer algebra approach is reflected in the presentation of the theory from a Lie transform approach. It is truly an algorithm in the sense of modern computer science: a clearly defined iterative procedure.

In this paper I would like to indicate the great genervality of the method by illustrating how it can be used to solve perturbation problems that are typically solved by other methods, often special ad hoc methods. In most cases I have chosen the simplest standard examples. In fact most of the paper consists of examples of problems that can be solved by Lie transforms, without spending too much time on the derivation or the theory. There are many topics of current research that are not considered here since this is to be an introduction not a summary of new results.