## Dynamic bifurcations: proceedings of a conference held in Luminy, France, March 5-10, 1990Dynamical Bifurcation Theory is concerned with the phenomenathat occur in one parameter families of dynamical systems(usually ordinary differential equations), when theparameter is a slowly varying function of time. During thelast decade these phenomena were observed and studied bymany mathematicians, both pure and applied, from eastern andwestern countries, using classical and nonstandard analysis.It is the purpose of this book to give an account of thesedevelopments. The first paper, by C. Lobry, is anintroduction: the reader will find here an explanation ofthe problems and some easy examples; this paper alsoexplains the role of each of the other paper within thevolume and their relationship to one another.CONTENTS: C. Lobry: Dynamic Bifurcations.- T. Erneux, E.L.Reiss, L.J. Holden, M. Georgiou: Slow Passage throughBifurcation and Limit Points. Asymptotic Theory andApplications.- M. Canalis-Durand: Formal Expansion of vander Pol Equation Canard Solutions are Gevrey.- V. Gautheron,E. Isambert: Finitely Differentiable Ducks and FiniteExpansions.- G. Wallet: Overstability in ArbitraryDimension.- F.Diener, M. Diener: Maximal Delay.- A.Fruchard: Existence of Bifurcation Delay: the DiscreteCase.- C. Baesens: Noise Effect on Dynamic Bifurcations:theCase of a Period-doubling Cascade.- E. Benoit: LinearDynamic Bifurcation with Noise.- A. Delcroix: A Tool for theLocal Study of Slow-fast Vector Fields: the Zoom.- S.N.Samborski: Rivers from the Point ofView of the QualitativeTheory.- F. Blais: Asymptotic Expansions of Rivers.-I.P.van den Berg: Macroscopic Rivers |

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### Contents

Dynamic Bifurcations | 1 |

Slow Passage Through Bifurcation and Limit Points | 7 |

Asymptotic Theory and Applications | 14 |

Copyright | |

11 other sections not shown

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abscissa adiabatic manifolds analytic function asymptotic expansion attracting behaviour bifurcation diagram bifurcation point canard curve canard orbit canard solutions change of variables class C1 coefficients computation consider corresponding critical point defined Definition delay denote Diener duck solution duck value dY/dX dynamic bifurcation e-expansion eigenvalues entrance-exit function entrance-exit related equivalent example exists a standard Exit(u exponentially small Figure finite function G give halo Hopf bifurcation hypothesis infinitely close infinitely large infinitely small infinitesimal initial condition interval invariant curve lemma limited linear logistic map magnifying glass matrix Morse point negative Newton polygon noise nonstandard nonstandard analysis obtain open set oscillations overstable solution period-doubling period-doubling bifurcation perturbation polynomial problem Proof of theorem properties Proposition prove random variable real function real number repelling satisfies semi resonant trajectory slow curve stable standard function standard neighbourhood stochastic sweep transform vector field zero