In the book of J. Jr. Palis and W. de Melo Geometric theory of dynamical systems: an introduction on the page $18$, there is a type of Poincaré-Bendixson theorem on $S^2$ or in plane or on cylinder as follows:
Theorem: Let $\mathcal M$ denotes the phase space, which may be the plane, cylinder, or two-sphere. $X \in \chi ^r(\mathcal M) $ ($r\ge 1$) be a vector field with a finite number of singularities. Take $p \in \mathcal M$ and let $\omega(p)$ be the $\omega$-limit set of $p$. Then one of the following possibilities holds
$(1)$ $\omega(p)$ is a singularity;
$(2)$ $\omega(p)$ is a closed orbit;
$(3)$ $\omega(p)$ consists of singularities $p_1, ... , p_n$ and regular orbits such that if $\gamma \in \omega(p)$ then $\alpha(\gamma) = p_i$, and $\omega(\gamma) = p_j$ for some $1\le i,j \le n$.
The proof uses from the Jordan Curve Theorem: every continuous closed curve without selfintersections separates $\mathcal M$ into two regions (two connected surface).
In the exercises of the end of the chapter, exercise $5$ the authors want to prove the following:
[Let $M^2$ be a 2-dimensional smooth manifold and] Let $F \subset M^2$ be a region homeomorphic to a Mobius band and let $X \in \chi^r(M^2)$ ($r\ge 1$) be a vector field such that $X_t(F) \subset F$ for all $t \ge 0$. If $X$ has a finite number of singularities in $F$ then the $\omega$-limit of the orbit of a point $p \in F$ either is a closed orbit or consists of singularities and regular orbits whose $\omega$ and $\alpha$-limits are singularities.
But on the Moebius strip, we can draw closed curves which have no “inside” or “outside”. Consider the curve which divides the strip in half, running halfway between the free edges. If we take a pair of scissors and cut along this curve, we will be left with a single connected surface. So, how we can prove the statement for the Moebius band?
