The Poincaré Bendixson theorem states:
If R is a closed bounded subset of $\mathbb{R}^{2}$ containing no fixed points and $\Psi_{t}\left ( x_{0} \right ) \in R$ for all $t\geq 0$, then, the omega-limit set $L_{\omega}\left ( x_{0} \right )$ is a periodic orbit.
Intuitively, either the trajectory is a closed orbit (which must contain a fixed point $x^{\ast}$ or it must spiral towards a closed orbit(or periodic orbit). In either case, it is understandable that it contains a fixed point $x^{\ast}$ in either case. But since R contains no fixed point, R is an annulus with the closed orbit being an orbit around the inner ring.
It comes across as odd to me. So in order to understand this, I reasoned that the only this can be true is if the closed orbit in R can orbit around a point S where S is a point in some other region
D $\subseteq \mathbb{R}^{2}$.
Or am I wrong? Can someone provide a more robust idea? Thanks in advance.