Topological characterizations of 2D gradient flows and 2D Morse-Smale flows and their generic intermediate flows

It is known that Morse-Smale vector fields form an open dense subset in the space of vector fields on orientable closed surfaces and are structurally stable. In particular, Morse vector fields have the same property in the space of gradient vector fields on such surfaces. In this talk, we characterize generic transitions on the space of 2D gradient flows and generic transitions on the space of Morse-Smale flows which are identified with "gradient flows with limit cycles". In fact, to describe these transitions, we characterize isolated singular points of gradient flows on surfaces. Using the characterization of such an isolated singular point, we topologically characterize gradient flows with finitely many singular points, Morse flows (i.e. Morse-Smale flows without limit cycles), and Morse-Smale flows on compact surfaces. In particular, a flow with finitely many singular points on a compact surface is gradient if and only if it is Morse-Smale-like without elliptic sectors or non-trivial circuits. Moreover, we describe generic transitions between Morse flows on the space of 2D gradient flows and those between Morse-Smale flows on the space of quasi-regular Morse-Smale-like flows. This talk is based on joint work with V. Kibkalo.