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It is known that $C^r$ Morse-Smale vector fields form an open dense subset inthe space of vector fields on orientable closed surfaces and are structurallystable for any $r \in \mathbb{Z}_{>0}$. In particular, $C^r$ Morse vectorfields (i.e. Morse-Smale vector fields without limit cycles) form an open densesubset in the space of $C^r$ gradient vector fields on orientable closedsurfaces and are structurally stable. Therefore generic time evaluations ofgradient flows on orientable closed surfaces (e.g. solutions of differentialequations) are described by alternating sequences of Morse flows andinstantaneous non-Morse gradient flows. To illustrate the generic transitions,we characterize and list all generic non-Morse gradient flows. To constructsuch characterizations, we characterize isolated singular points of gradientflows on surfaces. In fact, such a singular point is a non-trivial finitelysectored singular point without elliptic sectors. Moreover, consideringMorse-Smale flows as "generic gradient flows with limit cycles", wecharacterize and list all generic non-Morse-Smale flows.

Topological characterizations of Morse-Smale flows on surfaces and generic non-Morse-Smale flows