The $\omega$-limit set of a planar dynamical system is classified by Poincare-Bendixson theorem when it is compact, namely into three categories - an equilibrium point, a closed orbit, and finitely many equilibrium points with homoclinic and hetereoclinic orbits between them.
However, what happens when the $\omega$-limit set is noncompact? What are some good examples for such systems, and is there any good classification in that case? And is there some relatively general criterion to determine the compactness of the $\omega$-limit set?