Bifurcations of Planar Vector Fields Nilpotent Singularities and Abe
The book reports on recent work by the authors on the bifurcation structure of singular points of planar vector fields whose linear parts are nilpotent. The bifurcation diagrams of the most important codimension-three cases are studied in detail. The resu
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Lecture Nates in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen Subseries: lnstituto de Mathematics Pura e Aplicada Rio de Janeiro, Brazil (vol. 48) Adviser: C. Camacho
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F. Dumortier R. Roussarie J. Sotomayor H. Zoladek
Bifurcations of Planar Vector Fields Nilpotent Singularities and Abelian Integrals
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Authors Freddy Dumortier Limburgs Universitaire Centrum Universitaire Campus 3610 Diepenbeck, Belgium Robert Roussarie Departement de Mathematiques Universite de Bourgogne UFR de Sciences et Techniques Laboratoire de Topo1ogie (U. A. no. 755 du CNRS), B. P. 138 21004 Dijon, France Jorge Sotomayor Instituto de Maternatica Pura e Aplicada Estrada Dona Castorina 110 CEP 22460 Jardim Botanico Rio de Janeiro, Brazil Henryk Zoladek Institute of Mathematics Warsaw University 00-901 Warsaw, Poland
Mathematics Subject Classification (1991): 58F 14, 34C05, 34030
ISBN 3-540-54521-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54521-2 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany
Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 46/3140-543210 - Printed on acid-free paper
PREFACE
The study of bifurcations of families of dynamical systems defined by vector fields (i.e. ordinary differential equations) depending on real parameters is at present an active area of theoretical and applied research. Problems in mathematical biology, fluid dynamics, electrical engineering, among other applied disciplines, lead to multiparametric vector fields whose bifurcation analysis of equilibria (singular points) and oscillations (cycles) is required. The case of planar vector fields, due to the presence of regular as well as singular limit cycles is the first one, in increasing dimension of phase space, whose study cannot be fully reduced to the analysis of singularities and zeroes of algebraic equations, particularly when the number of parameters involved is larger than or equal to two. The results established in this volume illustrate the diversity of the algebraic, geometric and analytic methods used in the description of the variety of structural patterns that appear in the bifurcation diagrams of generic threeparameter families of planar vector fields, around singular points whose linear parts are nilpote
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