Trapping region - Proofs the existence of Limit cycle?

Question

Let us say I have a dynamical system: $$\dot r=f(r,\theta)$$ $$\dot \theta=g(r,\theta)$$ for which I can find a region such that trajectories only enter and never leave (taking the case where trajectories actually do enter). I know that from the Poincaré-Bendixson theorem that the trajectories in this region must either be closed or tend to a limit cycle (assuming no fixed points). I have however read that the above conditions imply the presence of a limit cycle. Thus my question is how do we know that we have a limit cycle instead of just closed orbits?

Answer 1

You wrote:

I know that from the Poincaré-Bendixson theorem that the trajectories in this region must either be closed or tend to a limit cycle (assuming no fixed points).

Clarifying a bit, the only thing that you can say is that:

In this region, assuming that there are no fixed points there is at least one closed trajectory.

It is easy to construct examples without limit cycles. On the other hand, sometimes you will be able to establish the existence of limit cycles using other additional techniques.