Dynamic Bifurcations: Proceedings of a Conference Held in Luminy, France, March 5-10, 1990, Issue 1493Eric Benoît 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. |
Contents
Dynamic Bifurcations | 1 |
Asymptotic Theory and Applications | 14 |
Formal Expansion of van der Pol Equation Canard Solutions are Gevrey | 29 |
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a₁ abscissa adiabatic manifolds analytic function appreciable asymptotic expansion bifurcation diagram buffer-point canard canard solutions change of variables class C¹ computation defined Definition delay denote Diener differential equation duck solution duck value dY/dX dynamic bifurcation eigenvalues Entr(u entrance-exit function example exists a standard Exit(u ɛ-expansion Figure finite formal function G halo Hopf bifurcation hypothesis infinitely close infinitely large infinitely small infinitesimal initial condition lemma limited linear logistic map macroscopic rivers magnifying glass Mathématiques matrix Morse point Newton polygon noise nonstandard nonstandard analysis oscillations overstable solution parameter period-doubling period-doubling bifurcation period-k perturbation polynomial problem Proof of theorem properties Proposition prove real number satisfies semi resonant trajectory singular slow curve slow-fast slowly varying stable standard function standard neighbourhood sweep Taylor expansion transform vector field zero αμ მყ