Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 32A / a / i / Solution

The Poincare-Bendixson theorem says that a nonempty compact $\omega$-limit set of a planar $C^1$ flow which contains only finitely many fixed points is either a fixed point, a periodic orbit, or a union of fixed points and connecting trajectories. In particular, if it contains no fixed point, it is a periodic orbit.