Dynamic Bifurcations Proceedings of a Conference held in Luminy, Fra

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1493

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen

1493

E. Benoit (Ed.)

Dynamic Bifurcations Proceedings of a Conference held in Luminy, France, March 5-10, 1990

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Editor Eric Benoit Centre de Mathematiques Appliquees Ecole des Mines 06565 Valbonne, France

Mathematics Subject Classification (1991): 34-02, 34Cxx, 34Exx, 34F05, 40H05, 4lA60

ISBN 3-540-54900-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54900-5 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 Dynamic Bifurcations Theory is concerned by the phenomena that occur in one parameter families of dynamical systems (usually ordinary differential equations), when the parameter is a slowly varying function of time. It turns out that during the last decade these phenomena were observed and studied by various mathematicians, both pure and applied, from eastern and western countries, using classical and nonstandard analysis. It is the purpose of this book to give an account of these developments. The first paper of C. Lobry is an introduction: the reader will find an explanation of the problems and some easy examples, he will understand also the right place of every paper in the book.

Contents Dynamic Bifurcations C. Lobry Slow Passage Through Bifurcation and Limit Points. Asymptotic Theory and Applications T. Erneux, E. L. Reiss, L. J. Holden and M. Georgiou

1

14

Formal Expansion of van der Pol Equation Canard Solutions are Gevrey M. Canalis-Durand

29

Finitely Differentiable Ducks and Finite Expansions V. Gautheron and E. Isambert

40

Overstability in Arbitrary Dimension G. Wallet

57

Maximal Delay F. Diener and M. Diener

71

Existence of Bifurcation Delay: the Discrete Case A. Fruchard

87

Noise Effect on Dynamic Bifurcations: the Case of a Period-doubling Cascade C. Baesens

107

Linear Dynamic Bifurcation with Noise E. Benoit

131

A Tool for the Local Study of Slow-fast Vector Fields: The Zoom A. Delcroix

151

Rivers from the Point of View of the Qualitative Theory S.N. Samborski

168

Asymptotic Expansions of Rivers F. Blais

181

Macroscopic Rivers LP. van den Berg

190

Dynamic Bifurcations Claude LOBRY U