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 many 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) $\omega(p)$ consists 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 drew the following picture in the blackboard:

enter image description here

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

So I assume that he asked that in the context given by the hypothesis of Poincaré-Bendixson theorem. Also he intended 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


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