Let $f:U\rightarrow\mathbb{R}^2$ a $C^1$ vector field in an open set $U\subseteq\mathbb{R}^2$ and $p\in U$ a regular point of $f$. Show that if $p\in \omega_p(f)$, then $\omega_p(f)$ is a periodic orbit of $f$.
I think... using the Poincaré–Bendixson theorem, it's enough to prove that $\omega_p(f)$ has no singular points, am I right? Any hints?
Hint: It seems it's much easier to argue using some standard lemmas leading to the Poincaré–Bendixson Theorem. First, since $p$ is regular, there is a flowbox around it. Further, since $f$ is continuous being regular for points is an open condition, that is to say by taking a small enough flowbox we may assume it does not contain a singular point. We also have an embedded compact interval $T$ passing through $p$ transverse to any orbit segment included in the flowbox. Since $p$ is recurrent (i.e. it's included in its own $\omega$-limit set) there is a time $t>0$ such that $\phi_{t}(p)$ is in $T$. If $\phi_{t}(p)$ is not $p$, then again by recurrence there must be another time $t'>t$ such that $\phi_{t'}(p)$ is in that segment of $T$ that is strictly between $\phi_{t}(p)$ and $p$, which is impossible.