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Perturbations singulières : approximations, stabilité pratique et applications à des modèles de compétition

Abstract : Chapter 1 recalls Tikhonov's theory for slow-fast systems in case of steady state fast dynamics. Chapter 2 deals with Pontryagin-Rodygin's theorem where the fast dynamics are periodic. We propose a new proof of this theorem emphasizing on its topological features. These results concern bounded time intervals. We indicate in Chapter 3 how the geometrical theory of perturbations treats the case of the oscillating fast dynamics. In Chapter 4, results for unbounded time intervals are established when the fast dynamics converge to a positively invariant compact subset. These results lead to practical stability theorems. Chapter 5 is devoted to the case where the fast equation has cycles with relaxation. A rigorous result describes the slow motion, the proof being based on the stroboscopic method. The theorems are stated in terms of the classical mathematics and proved with the use on non standard analysis tools. Chapter 6 is a study of a fourth-dimensional competition model. Some talks given by Pr. C. Lobry represent the starting point of this work. C. Lobry built a model of 3 species x1, x2 and x3 competing on one single prey specie s, the coexistence of x2 and x3 seeming possible through some numerical simulations, while s and x1 oscillate. We determine the averaged system which describes the slow motion of the couple (x2,x3). We establish sufficient conditions of persistence and we illustrate the results with examples and numerical simulations.
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Contributor : Karim Yadi <>
Submitted on : Thursday, August 27, 2009 - 5:14:12 PM
Last modification on : Tuesday, October 16, 2018 - 2:26:02 PM
Long-term archiving on: : Saturday, November 26, 2016 - 11:29:46 AM


  • HAL Id : tel-00411503, version 1



Karim Yadi. Perturbations singulières : approximations, stabilité pratique et applications à des modèles de compétition. Mathématiques [math]. Université de Haute Alsace - Mulhouse, 2008. Français. ⟨tel-00411503⟩



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