We have and ode $x'=f(x)$, where $x \in \mathbb{R}^2$ and $f: \mathbb{R}^2\to \mathbb{R}^2$ a $\mathcal{C}^1$ function.
Suppose that $\gamma$ is a periodic orbit for this ode.
Suppose also that there exists $y \in \mathbb{R}^2 $, $y \notin \gamma$, such that $\omega(y) = \gamma$ (it means that $\gamma$ is a limit cycle).
Prove that there exist an open set U in $\mathbb{R}^2$ such that $\forall z \in U$, $\omega(z) = \gamma$.
I tried to use Poincaré Bendixson theorem and continuity of solutions with respect to initial conditions in order to prove that such $U$ exists. I couldn't see a clear way to do this.
Any ideas?