We consider a \({\mathcal {C}}^1\) 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 define non empty extended limit sets contained in the boundary of U. We give an elementary proof of this result, independent of the classical Poincaré–Bendixson theorem. A trapping triangle \({\mathcal {T}}\) based at p, for a \({\mathcal {C}}^1\) vector field X defined on an open subset \({\mathcal {U}}\) of the plane, is a topological triangle with a corner at a point p located on the boundary \(\partial {\mathcal {U}}\) 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 on the properties of X along \(\partial {\mathcal {U}}\). For instance, X could be non differentiable at p, as in the example presented in the last section.

我们考虑在平面的开子集U上定义的\({\mathcal {C}}^1\)向量场X。如果X没有奇异点，如果ü简单地连接时，庞加莱-本迪克松定理的一个弱版本说，极限集X在ü是空的，但一个可以定义包含在边界非空的扩展极限集ü。我们给出了这个结果的基本证明，独立于经典的庞加莱-本迪克森定理。一个陷阱三角形\({\mathcal {T}}\)基于p，对于\({\mathcal {C}}^1\)向量场X定义在平面的开子集\({\mathcal {U}}\)上，是一个拓扑三角形，其角位于边界\(\partial {\mathcal {U}}\)上的点p和一个很好地控制了X沿边的横向。弱 Poincaré-Bendixson 定理的主要应用是p处的陷阱三角形包含一个向p点收敛的分离线。这不取决于X沿\(\partial {\mathcal {U}}\)的属性。例如，X可能在p处不可微，如上一节中的示例所示。