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Dynamic bifurcations. Proceedings of a conference, held in Luminy, France, March 5-10, 1990. (English) Zbl 0756.34042

Lecture Notes in Mathematics. 1493. Berlin etc.: Springer-Verlag. 219 p. (1991).

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[The articles of this volume will not be indexed individually.]
This volume contains a series of articles discussing various aspects of dynamic bifurcation. This theory deals with the effects arising if one considers the bifurcation parameter as varying slowly in time. This is in contrast to standard bifurcation theory, where for each value of the parameter the system (experiment) is considered to last forever.
This book is an attempt to present the different articles in a unified manner. One common feature is the language of nonstandard analysis. In most articles this language is used such that it can be understood heuristically. However it seems that for a complete understanding of all articles some knowledge of nonstandard analysis is necessary.
In the first article by C. Lobry, titled Dynamic bifurcations, the fundamental questions are presented, some of the main features are illustrated by means of simple examples. Especially the phenomenon of the delay of bifurcation is explained nicely. This article puts all the articles in the volume in perspective. The relation to the main questions is illustrated. Therefore there is no need of discussing the contributions individually. Let me just mention the contributors and their titles: T. Erneux, E. L. Reiss, L. J. Holden and M. Georgiu: Slow passage through bifurcation and limit points. Asymptotic theory and applications; M. Canalis-Durand: Formal expansion of van der Pol equation canard solution are Gevrey; V. Gautheron and E. Isambert: Finitely differentiable ducks and finite expansions; G. Wallet: Overstability in arbitrary dimension; F. Diener and M. Diener: Maximal delay; A. Fruchard: Existence of bifurcation delay: the discrete case; C. Baesens: Noise effect on dynamic bifurcation: the case of a period doubling cascade; E. Benoît: Linear dynamic bifurcation with noise; A. Delcroix: A tool for the local study of slow fact vector fields: the zoom; S. N. Samborski: Rivers from the point of view of the qualitative theory; F. Blais: Asymptotic expansions of rivers; I. P. van den Berg: Macroscopic rivers.

MSC:

34C23 Bifurcation theory for ordinary differential equations
37G99 Local and nonlocal bifurcation theory for dynamical systems
00B25 Proceedings of conferences of miscellaneous specific interest
34-06 Proceedings, conferences, collections, etc. pertaining to ordinary differential equations
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