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Contribution à l'Etude de la Bifurcation de Hopf dans le Cadre des Equations Différentielles à Retard, Application à un Problème en Dynamique de Population.

Abstract : Our first goal in this work is to give a proof of exchange of stability from the trivial
branch to the bifurcated one. This proof is based on the two following steps:
i) reduction of the equation to a two-dimensional system via the variation of constant
formula end the center manifold theorem.
ii) Estimation of the distance between solutions of the original equation and the bifurcated
periodic solutions.
We obtain an estimate of the stability region.
The second goal is to study the dynamics of Haematopoietic Stem Cells (HSC) Model
with one delay.
The model, was initially introduced by Mackey (1978). There are two possible stationary
states. One of them is trivial and unstable, the second is nontrivial, depending on
the delay \tau.
We prove the existence of a critical value ¿0 of the delay \tau in which the exchange of
stability of nontrivial stationary state may occur.
We introduce also an approachable model depending on this critical value of the delay,
such that the nontrivial stationary state do not depend on the delay which is the same
one of Mackey model at \tau =\tau_{0}.
By a similar study of the approachable model as in Mackey model, we obtain the existence
of the bifurcated periodic solution branch around the nontrivial stationary state.
In the end, we give an explicit algorithm for calculating the elements of bifurcation.
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https://tel.archives-ouvertes.fr/tel-00076961
Contributor : Radouane Yafia <>
Submitted on : Saturday, June 3, 2006 - 11:56:58 AM
Last modification on : Sunday, April 12, 2020 - 10:26:02 PM
Long-term archiving on: : Monday, September 17, 2012 - 3:00:17 PM

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Radouane Yafia. Contribution à l'Etude de la Bifurcation de Hopf dans le Cadre des Equations Différentielles à Retard, Application à un Problème en Dynamique de Population.. Mathématiques [math]. Faculte des Sciences El Jadida, 2005. Français. ⟨tel-00076961⟩

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