Moscow University Mathematics Bulletin, 46 (1991) 3, 54-55. Reviews: MR 93k:58196, ZM 785.54039.
The famous Poincare-Bendixson Theorem states that under suitable conditions the limit set of a trajectory of a differential equation with uniqueness in the plane is a cycle:
(There are versions of this theorem for flows on surfaces and even for codimension 1 foliations). It was generalized by V. V. Filippov to the limit set of a sequence of trajectories without uniqueness provided there are no intersections or self-intersections. The proof is purely topological. In this note we extend it further to include the following result: the limit set of a sequence of cycles is a cycle.