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Title: | Théorème de Poincaré‐Bendixson et applications |
Authors: | Bahayou, Mohamed-Amine Bakria, Ahmed |
Keywords: | Ensemble w‐limite orbite périodique redressement d’un champ de vecteurs |
Issue Date: | 2019 |
Publisher: | UNIVERSITÉ KASDI MERBAH OUARGLA |
Abstract: | Le problème de l’existence d’orbites périodiques et de points critiques est fonda‐
mental dans l’analyse du comportement des équations différentielles et dans plusieurs
applications. Cependant, dans de nombreux cas, il n’est pas facile de trouver une
telle solution. Les systèmes à deux dimensions jouent ici un rôle important. L’une
des raisons est qu’une équation unidimensionnelle du second ordre peut être ré‐
duite à un système de deux équations du premier ordre. Le Théorème de Poincaré‐
Bendixson donne des conditions permettant de prouver l’existence d’une solution
périodique de l’équation. De plus, dans de nombreux cas, pour les systèmes à deux
dimensions, l’existence d’une orbite périodique donne également l’existence d’un
point critique. L’existence d’un type particulier d’orbites périodiques, c’est‐à‐dire
de cycles limites, est particulièrement intéressante. Elle découle fréquemment du
Théorème de Poincaré‐BendixsonAbstract
The problem of existence of periodic orbits and critical points is fundamental in
the analysis of behaviour of differential equations and in several applications. How‐
ever, in many cases it is not easy to fnd such a solution. Two‐dimensional systems
play here an important role. One of reasons is that a one‐dimensional equation of
the second order may be reduced to a system of two equations of the frst order.
The Poincaré–Bendixson Theorem gives conditions which enable us to prove the
existence of a periodic solution of the equation. Moreover, for two‐dimensional sys‐
tems in many cases the existence of a periodic orbit gives also the existence of a
critical point. The existence of a particular kind of periodic orbits, i.e. limit cycles, is
in many situations particularly interesting. Frequently, it follows from the Poincaré–
Bendixson Theorem. The problem of existence of periodic orbits and critical points is fundamental in the analysis of behaviour of differential equations and in several applications. How ever, in many cases it is not easy to find such a solution. Two‐dimensional systemsplay here an important role. One of reasons is that a one‐dimensional equation ofthe second order may be reduced to a system of two equations of the first order. The Poincaré–Bendixson Theorem gives conditions which enable us to prove the existence of a periodic solution of the equation. Moreover, for two‐dimensional systems in many cases the existence of a periodic orbit gives also the existence of acritical point. The existence of a particular kind of periodic orbits, i.e. limit cycles, is in many situations particularly interesting. Frequently, it follows from the PoincaréBendixson Theorem. |
Description: | ANALYSE FONCTIONNELLE |
URI: | http://dspace.univ-ouargla.dz/jspui/handle/123456789/21774 |
Appears in Collections: | Département de Mathématiques - Master |
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File | Description | Size | Format | |
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bakria-ahmed.pdf | 773,35 kB | Adobe PDF | View/Open |
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