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

Abstract

<p style='text-indent:20px;'>It is known that <inline-formula><tex-math id="M1">\begin{document}$ C^r $\end{document}</tex-math></inline-formula> Morse-Smale vector fields form an open dense subset in the space of vector fields on orientable closed surfaces and are structurally stable for any <inline-formula><tex-math id="M2">\begin{document}$ r \in \mathbb{Z}_{\geq 1} $\end{document}</tex-math></inline-formula>. In particular, <inline-formula><tex-math id="M3">\begin{document}$ C^r $\end{document}</tex-math></inline-formula> Morse vector fields (i.e. Morse-Smale vector fields without limit cycles) form an open dense subset in the space of <inline-formula><tex-math id="M4">\begin{document}$ C^r $\end{document}</tex-math></inline-formula> 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 (e.g. bifurcations of singular points, transitions via heteroclinic separatrices), 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.</p>

Combinatorial structures of the space of gradient vector fields on compact surfaces

“…Lemma 3.3 and Lemma 3.4 imply Theorem 3.1. Notice that the similar statement holds for quasi-regular Morse-Smale-like flows under the non-existence of fake limit cycles because quasi-regular Morse-Smale-like flows are quasi-Morse-Smale (i.e the resulting flows obtained from quasi-regular gradient flows by replacing singular points with limit cycles and pasting limit cycles) (see [6] for definition details). We will state the statement in the second from the last section more precisely.…”

“…(2) There are at most finitely many limit cycles. In [6,Theorem B], it is shown that a flow on a compact surface is a gradient flow with finitely many singular points if and only if the flow is Morse-Smale-like without elliptic sectors or non-trivial circuits.…”

“…1 Though there are typos in [6,Lemma 7.7], the correctly modified statement is contained in these theorems.…”

“…By these facts, one characterizes a "generic" non-Morse gradient flow on a compact surface to describe a generic time evaluation of gradient flows on orientable compact surfaces (e.g. solutions of differential equations) which is an alternating sequence of Morse flows and instantaneous ⇦ ⇦ non-Morse gradient flows under no physical or symmetric restrictions [6]. On the other hand, "non-generic" intermediate vector fields (e.g.…”