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Some Applications of the Poincaré-Bendixson Theorem

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We consider a C 1 vector field X defined on an open subset U of the plane, with compact closure. If X has no singular points and if U is simply connected, a weak version of the Poincaré-Bendixson Theorem says that the limit sets of X in U are empty but that one can defined non empty extended limit sets contained into the boundary of U. We give an elementary proof of this result, independent of the classical Poincaré-Bendixson Theorem. A trapping triangle T based at p, for a C 1 vector field X defined on an open subset U of the plane, is a topological triangle with a corner at a point p located on the boundary ∂U and a good control of the tranversality of X along the sides. The principal application of the weak Poincaré-Bendixson Theorem is that a trapping triangle at p contains a separatrix converging toward the point p. This does not depend of the properties of X along ∂U. For instance, X could be non differentiable at p, as in the example presented in the last section.