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

Claude Lobry (auth.), Eric Benoît (eds.)9780387549002, 0387549005, 3540549005

Dynamical Bifurcation Theory is concerned with 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. During the last decade these phenomena were observed and studied by many 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, by C. Lobry, is an introduction: the reader will find here an explanation of the problems and some easy examples; this paper also explains the role of each of the other paper within the volume and their relationship to one another. CONTENTS: C. Lobry: Dynamic Bifurcations.- T. Erneux, E.L. Reiss, L.J. Holden, M. Georgiou: Slow Passage through Bifurcation and Limit Points. Asymptotic Theory and Applications.- M. Canalis-Durand: Formal Expansion of van der Pol Equation Canard Solutions are Gevrey.- V. Gautheron, E. Isambert: Finitely Differentiable Ducks and Finite Expansions.- G. Wallet: Overstability in Arbitrary Dimension.- F.Diener, M. Diener: Maximal Delay.- A. Fruchard: Existence of Bifurcation Delay: the Discrete Case.- C. Baesens: Noise Effect on Dynamic Bifurcations:the Case of a Period-doubling Cascade.- E. Benoit: Linear Dynamic Bifurcation with Noise.- A. Delcroix: A Tool for the Local Study of Slow-fast Vector Fields: the Zoom.- S.N. Samborski: Rivers from the Point ofView of the Qualitative Theory.- F. Blais: Asymptotic Expansions of Rivers.-I.P. van den Berg: Macroscopic Rivers

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Table of contents :
Dynamic bifurcations….Pages 1-13
Slow passage through bifurcation and limit points. Asymptotic theory and applications….Pages 14-28
Formal expansion of van der pol equation canard solutions are gevrey….Pages 29-39
Finitely differentiable ducks and finite expansions….Pages 40-56
Overstability in arbitrary dimension….Pages 57-70
Maximal delay….Pages 71-86
Existence of bifurcation delay: The discrete case….Pages 87-106
Noise effect on dynamic bifurcations: The case of a period-doubling cascade….Pages 107-130
Linear dynamic bifurcation with noise….Pages 131-150
A tool for the local study of slow-fast vector fields: The zoom….Pages 151-167
Rivers from the point of view of the qualitative theory….Pages 168-180
Asymptotic expansions of rivers….Pages 181-189
Macroscopic rivers….Pages 190-209