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.

Introduction to resurgence theory

This section is a very brief introduction to resurgence. Forgetting the purely technical difficulties, our aim is to present the noteworthy simple basic ideas of the Ecalle theory. In this way, we shall restrict ourselves to the quite simple algebra of simple resurgent functions, which gives a very pleasant context for beginning the theory.

The framework is the following. We begin by defining a subalgebra of the multiplicative algebra of formal power series C[[x-1]], furnishing through the Borel transformation a convolutive subalgebra of analytic germs at the origin. On the other hand, in order to sum by a Laplace transformation, the analytic continuation of these germs must have only “few” singularities, a notion which has to be stable under the convolution product. After having defined the algebra of simple resurgent functions, we get naturally the notion of resurgent symbols by a comparison of the different summations, in other words by an analysis of the Stokes phenomena. These can be described either with the help of an automorphism of algebra or with new differentiations, the alien differentiations.

A bibliography will allow the reader to go further into the theory. At first, we naturally send the reader to the whole work of Ecalle himself. We have followed here more or less the clear presentation of the reference [CNP] where one can find all the basic tools with complete proofs and some applications.

Abstract

This work explores various general properties of systems of nonlinear ordinary differential equations obtained by using nonlinear transformations: further reductions of Poincaré normal forms for autonomous and non-autonomous systems, Lotka-Volterra universal standard format and generalization of Painlevé singularity analysis for detecting integrable systems. These methods are progressively implemented in the computer algebra program NODES.

Introduction

The main purpose of this work is to present some general results for systems of nonlinear ODE's obtained through quasi-monomial transformations. These transformations (discovered independently by a mathematician: A. Br'uno [1]; engineers: M. Peschel and W. Mende [2]; and physicists: the authors of the present article [3, 4, 5]), provide a powerful algebraic computational scheme for the analysis of nonlinear ODE's.

Essentially, three types of results are obtained in this framework:

I. Direct decoupling and/or integrability conditions under quasi-monomial transformations (QMT) with explicit closed-form construction of the reduced ODE systems and first integrals.

II. Reduction through QM-transformation to the Lotka-Volterra standard format.

III. Extension of the Painlevé test for integrability.

- Results of type I are based on a general matrix representation of systems of ODEs with polynomial nonlinearities closely associated to the QM transformations. Decoupling and integrability conditions arise from singularities of the matrices involved in this representation [5]. Such a singularity is generic in Poincaré-Dulac normal forms. As a consequence, the dimensions of these systems are reduced after a well-defined QM transformation.

This book corresponds to a graduate course which has been taught many times at the universities Paris-Sud, Paris-Nord and Paris 7, and other places. The aim of this text is to give the foundation of what is nowadays called microlocal analysis in the C∞ framework, as it was created in the sixties and seventies by Kohn-Nirenberg, Maslov and Hörmander. Our presentation follows essentially the one given by Hörmander [Hö2]; as for the symplectic geometry, we have been inspired by the lecture notes of Duistermaat [D].

This subject is of growing importance, with a range of applications going beyond the original problems of linear partial differential equations. In particular the link with quantum mechanics is now firmly established, and there is a growing number of books covering more or less specialized parts of the theory. We believe that a short monograph concentrating on the basic principles could be of value, not only for the graduate student, but also for the mathematician who wants to get quickly into the subject and to understand its basic mechanisms. For this, the classical PDE framework seemed to us the most suitable one. The basic principles of microlocal analysis are essentially only two: integration by parts and the method of stationary phase. Compared with the article [HÖ2] and many of the other presentations, we have insisted even more on the stationary phase method, which appears already in the development of the theory of pseudodifferential operators and also (as in Melin-Sjöstrand [MS]) in the proof of the equivalence of phase functions in the global theory of Fourier integral operators.

When the Organizing Commitee of CADE began to choose the program of CADE-92, it was decided that D-modules would be a central topic at this conference.

The theory of D-modules is quite recent. It began in the late sixties and at first was considered to be quite abstract and difficult. Over the years the situation improved with the development of the theory and its applications. The organizers felt that it was time to try to introduce it to a larger audience interested in differential equations and computer algebra, since the theory of D-modules offers an excellent way to effectively handle linear systems of analytic PDEs.

Once this decision was made it was natural to ask Bernard Malgrange to be the “invité d'honneur” at CADE-92, with the task of lecturing about D-modules in a way adapted to an audience interested in effectivity. This was natural because Bernard Malgrange is not only one of the most famous mathematicians in this field, but also because he is perhaps the true originator of this direction. It is generally admitted that D-module theory began in the early seventies with the fundamental work of I. N. Berstein and of the Japanese school around M. Sato, but in fact Bernard Malgrange introduced the basic concepts ten years ago for the constant coefficients case (see his 1962 Bourbaki report “systèmes différentiels à coefficients constants”), and later for the general case (see his lectures at Orsay Cohomologie de Spencer (d'après Quillen)).