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

TL;DR

This paper characterizes and lists all generic non-Morse gradient and Morse-Smale flows on surfaces, highlighting the transitions between Morse and non-Morse flows in the context of structural stability and generic properties.

Contribution

It provides a complete characterization of generic non-Morse gradient flows and non-Morse-Smale flows on surfaces, expanding understanding of flow transitions and singular point structures.

Findings

Characterized isolated singular points of gradient flows on surfaces.

Listed all generic non-Morse gradient flows.

Listed all generic non-Morse-Smale flows.

Abstract

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