If a system in $\mathbb{R}^2$ has no periodic orbit, and has only one critical point which is asymptotically stable, prove that any bounded orbit $(x(t):t>0)$ must converge to the unique critical point as $t \rightarrow \infty$.
I believe it is an application of the Poincare-bendixson Theorem, but I cannot seem to connect the dots. Any help?