Théorème de Poincaré‐Bendixson et applications

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.