Last class we were proving Poincaré-Bendixson theorem in $ \mathbb R^2$ which states that:

Assume that the positive orbit $ \mathcal O^+(p)$ is contained in a compact subset $ K$ of the planar domain $ D$ of the differential equation $ x’=X(x)$ . Assume further that $ X$ has only finitely maney fixed points in $ K$ . Then one of the following is satisfied:

a) $ \omega(p)$ is a periodic orbit;

b) $ \omega(p)$ is a single fixed point

c) $ \omega(p)$ consistis of a finite number of fixed points, together with a finite set of orbits such that for each orbit its $ \alpha$ -limit set is a single fixed point and its $ \omega$ -limit set is also a single fixed point.

During the proof, he stopped the class and drawed the following picture in the black board:

and asked, can $ \omega(p)$ be like that?

So I assume that he asked that in the context given by the hyphotesis of Poincaré-Bendixson theorem. Also he intented to picture with those dots the singularities of the field. I want to say that the answer is no, because I think that every singularity must be connected by an orbit, which does not happen in this picture. But the problem is: I can’t justify that. I’ve read the poincaré-bendixson demonstration quite a few times, but I can’t find this justificative there. Any insight would be very helpful. Thank you