I have a quick question about the Poincare-Bendixson Theorem in 2D. The way I understand the theorem is that, loosely, if we have some region $R$ that is bounded, connected, and closed, and contains no fixed points, then if any trajectory that enters $R$ remains in $R$ as $t\rightarrow\infty$ then $R$ must contain a limit cycle.
My question is about the applicability of the theorem; I have recently done a problem which had an unstable fixed point at the origin, a stable limit cycle at $r=\sqrt{2}$, and an unstable limit cycle at $r=\sqrt{7}$
Using the P-B Theorem in an annulus between $r=1$ and $r=2$ I can easily show the existence of the first limit cycle, since $\dot{r}$ changes sign at either boundary and therefore any trajectories which enter must remain within the annulus. So far so good.
However, when I came to analyse the second limit cycle, because it is unstable I found that no trajectories entered the region, but there was a net flow out of the annulus from both sides.
My question is two fold:
1) Is the P-B Theorem not applicable in this case? I wondered if it was a technicality that because no trajectories entered, there were no trajectories to leave HAVING ALREADY entered, but I feel like the answer to this is probably an emphatic no. It may also be that my understanding of the Theorem isn't rigorous or expansive enough, in which case I would love to know what I'm missing.
2) I deduced that there must be a limit cycle in the region anyways, since $\dot{r}$ changed signs in a similar way to the previous case, just in the "opposite order", indicating an unstable limit cycle.
The problem is that in the problem itself $\dot{r}$ was easily factorised so I knew where the limit cycles were anyways, which sort of made me think this was a case of "obvious in hindsight".
