Dynamic Bifurcations: Proceedings of a Conference held in Luminy, France, March 5-10, 1990
Springer Berlin Heidelberg, Nov 13, 1991 - Mathematics - 222 pages
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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Slow Passage Through Bifurcation and Limit Points
Asymptotic Theory and Applications
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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