I believe that the answer is yes (provided there are no further limit cycles within the first one).
Let there be a stable limit cycle with no other limit cycles within it. By reversing time, $t \rightarrow -t$, we get a new system with trajectories bounded to the interior of the limit cycle. Using, say, Brouwer’s fixed point theorem, one can show that there must be a fixed point within the region.
However, when reading different formulations of the Poincaré–Bendixson theorem, I often encountered the statements like this:
If the trajectories are bounded and there are no fixed points, then the trajectories must converge to a limit cycle.
Are these are just unlucky/incorrect formulations or there is something in it?
Best Answer
Well, if the trajectories are confined to some region which does contain a fixed point, then it may happen that they all converge to that fixed point, so that there are no limit cycles in the region. So by assuming that there are no fixed points, you rule out that possibility.
But note that when you apply Poincaré–Bendixson, the trapping region is usually an annulus (or something topologically equivalent to that), so you don't consider the whole region inside the potential limit cycle. As you say, there is necessarily a fixed point inside the inner circle of the annulus, but that point doesn't belong to the annulus.