Abstract

The famous Poincaré-Bendixson theorem is one of the key results of the qualitative theory of continuous bidimensional dynamical systems. It states that if the omega-limit set (the set of limit points) of an orbit of a continuous flow on the sphere contains no stationary points then this set is a periodic orbit (thus homeomorphic to the circle). Starting from the original results by Poincaré (1886) and Bendixson (1901) we survey the main developments in the area and present some recent work by G. Soler López and the speaker providing a full topological characterization of omega-limit sets for continuous flows on compact closed surfaces.