IRMA Lectures in Mathematics and Theoretical Physics 3 Edited by Vladimir G. Turaev Institut de Recherche Mathe´matique Avance´e Universite´ Louis Pasteur et CNRS 7 rue Rene´ Descartes 67084 Strasbourg Cedex France IRMA Lectures in Mathematics and Theoretical Physics 1 Deformation Quantization, Gilles Halbout (Ed.) 2 Locally Compact Quantum Groups and Groupoids, Leonid Vainerman (Ed.) From Combinatorics to Dynamical Systems Journe´es de Calcul Formel Strasbourg, March 22Ϫ23, 2002 Editors F. Fauvet C. Mitschi ≥ Walter de Gruyter · Berlin · New York Editors Fre´de´ric Fauvet, UFR de Maths et IRMA Ϫ CNRS UMR 7501, Universite´ Louis Pasteur, 7, rue Rene´ Descartes, 67084 Strasbourg Cedex, France, e-mail: [email protected] Claude Mitschi UFR de Maths et IRMA Ϫ CNRS UMR 7501, Universite´ Louis Pasteur, 7, rue Rene´ Descartes, 67084 Strasbourg Cedex, France, e-mail: [email protected] Series Editor Vladimir G. Turaev Institut de Recherche Mathe´matique Avance´e (IRMA), Universite´ Louis Pasteur Ϫ C.N.R.S., 7, rue Rene´ Descartes, 67084 Strasbourg Cedex, France, e-mail: [email protected] Mathematics Subject Classification 2000: 11-04; 05E99, 11M99, 12H05, 12H10, 32C38, 33E05, 34A25, 34A30, 34A34, C20, 34E05, 34E10, 34M25, 34M30, 34M35, 40A05, 40D05, 68W30, 70F07, 70H05, 70H06 Physics and Astronomy Classification SchemeϪ1995: 02.30.Ϫf, 0260.Ϫx, 02.70.Ϫc, 05.45.ϩb, 47.27.Ϫi, 82.40.Ϫg Key words: Multiple zeta values, free Lie algebras, quasi-symmetric functions, irregular singularities, normal forms, mero- morphic connections, ordinary differential equations, Gevrey asymptoticity, differential Galois theory, Lokta- Volterra systems, equation of Kuramoto and Sivashinsky, unidimensional cubic complex equation of Ginzburg and Landau, hybrid automaton and modeling, optimal control, nonlinear ordinary differential equations. Țȍ Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress Cataloging-in-Publication Data From combinatorics to dynamical systems : journe´es de calcul formel, Strasbourg, March 22Ϫ23, 2002 / editors F. Fauvet, C. Mitschi. p. cm. Ϫ (IRMA lectures in mathematics and theoretical physics ; 3) Includes bibliographical references. ISBN 3-11-017875-3 (alk. paper) 1. Algebra Ϫ Data Processing Ϫ Congresses. I. Fauvet, F. (Fre´de´- ric). II. Mitschi, C. (Claude). III. Series. QA155.7.E4F76 2003 512Ϫdc22 2003062471 ISBN 3-11-017875-3 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at Ͻhttp://dnb.ddb.deϾ. Ą Copyright 2003 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be repro- duced in any form or by any means, electronic or mechanical, including photocopy, recording, or any informa- tion storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Cover design: I. Zimmermann, Freiburg. Depicted on the cover is the Strasbourg Cathedral. Typeset using the authors’ TEX files: I. Zimmermann, Freiburg. Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen. Preface This volume presents the proceedings of the Conference “Journées de calcul formel en l’honneur de Jean Thomann”, which took place at IRMA in March 2002. This meeting was aimed at bringing together mathematicians working in various areas of Thomann’s interests, specifically in dynamical systems, computer algebra, and theoretical physics. Jean Thomann was born on June 30, 1936, in Strasbourg. He received his math- ematical education in Strasbourg where among his professors were Marcel Berger, Jean-Louis Koszul and René Thom. While a research engineer with the IN2P3, In- stitut National de Physique des Particules, he worked at the Institut de Recherche Nucléaire, at the Centre de Calcul du CNRS in Strasbourg, and at IRMA. It is important to mention that the present volume has historical ties to the proceed- ings(1), (2), (3)ofthree pioneering workshops “Computer Algebra and Differential Equations” (CADE) held in Grenoble 1988, Cornell 1990, Luminy 1992. These pro- ceedings have become standard references in the area, and as with these classical texts, this volume includes papers of both theoretical and applied nature. The unifying theme of these proceedings is computer algebra. As for the orga- nization of the material, the book begins with articles of combinatorial or algebraic nature, then proceeds to articles involving more analytical tools. The book includes a foreword by Jean-Pierre Ramis (in French) followed by nine articles. The first paper by M. Espie, J.-C. Novelli and G. Racinet is devoted to compu- tations in certain graded Lie algebras related to multiple zeta values. Some of these computations have been implemented in computer algebra systems with an unexpected experimental result leading to a new conjecture. The next four articles study differ- ential systems with irregular singularities from algebraic as well as analytic points of view. The article of E. Corel gives an algebraic and algorithmic treatment of for- mal exponents at an irregular singularity and provides new and effective methods for computing such exponents and exponential parts for solutions of linear differential systems. The article of J. L. Martins studies irregular meromorphic connections in higher dimensions based on the seminal work of C. Sabbah. Using essentially alge- braic techniques, the author obtains results concerning the summability of solutions. The paper of M. A. Barkatou, F. Chyzak and M. Loday-Richaud describes and com- pares various algorithms concerning the rank reduction of differential systems. It is 1Computer Algebra and Differential Equations, edited by E. Tournier, Academic Press, 1989 2Differential Equations and Computer Algebra, edited by M. F. Singer, Academic Press, 1991 3Computer Algebra and Differential Equations, London Math. Soc. Lecture Note Ser. 193, edited by E. Tournier, Cambridge University Press, 1994 vi Preface intimately related to questions of classification of irregular connections and compu- tations of differential Galois groups. The article of M. Canalis-Durand presents new formal and numerical algorithms providing estimates of the growth of coefficients for a certain type of divergent series which are ubiquitous in dynamical systems with irregular singularities, such as in Quantum Field Theory. The next three papers contain effective calculations, formal and numerical, for dy- namical systems arising from certain questions in physics. The article of D. Boucher and J.-A. Weil presents computations of non-integrability of Hamiltonian systems, based on the Morales-Ramis theorem in differential Galois theory. The paper of L. Brenig uses the Quasi Polynomial formalism and generalized Lotka–Volterra nor- mal forms to obtain effective integrability for some dynamical systems, with interesting combinatorial consequences. The article of R. Conte, M. Musette and T.-L. Yee in- volves computations of Padé approximants related to concepts of analytic integrability of equations of physical significance with chaotic behaviour. The article of J. Della Dora and M. Mirica-Ruse explains a new mathematical formalization of the recent concept of hybrid systems, with ideas and techniques from dynamical systems and automata theory. This conference was supported by IRMA (Strasbourg), with additional financing by Institut IMAG (Grenoble), Institut Universitaire de France and Université d’Angers, which we thankfully acknowledge. We express our gratitude to Jean-Pierre Ramis for his constant support, to the director and the staff of IRMA for their help during the Conference, to Vladimir Turaev, editor of the IRMA Lectures in Mathematics and Theoretical Physics for kindly suggesting to publish these proceedings, and to the mathematics editor of de Gruyter. Strasbourg, September 2003 Frédéric Fauvet and Claude Mitschi Table of Contents Preface ....................................................................v Hommage à Jean Thomann by Jean-Pierre Ramis .................................ix Marc Espie, Jean-Christophe Novelli, and Georges Racinet Formal computations about multiple zeta values................................1 Eduardo Corel Algorithmic computation of exponents for linear differential systems ............17 José Luis R. F. Martins Sur les représentations de solutions formelles d’équations différentielles.........63 Moulay Barkatou, Frédéric Chyzak et Michèle Loday-Richaud Remarques algorithmiques liées au rang d’un opérateur différentiel linéaire ......87 Mireille Canalis-Durand Gevrey normal form of systems of differential equations with a nilpotent linear part ................................................................131 Delphine Boucher and Jacques-Arthur Weil Application of J.-J. Morales and J.-P. Ramis’ theorem to the planar three-body problem .......................................................163 Léon Brenig On a constructive approach to the solution of dynamical systems ..............179 T.-L. Yee, R. Conte et M. Musette Sur la “solution analytique générale” d’une équation différentielle chaotique....195 Jean Della Dora and Mihaela Mirica-Ruse Hybrid Systems ..........................................................213 Jean Thomann Hommage à Jean Thomann En préambule à ce volume d’hommage à Jean Thomann, je voudrais évoquer quelques souvenirs de notre aventure