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

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.

Along a periodic orbit the vector field is tangent and makes one full turn, so its index around the orbit is $+1$. If the enclosed region contained no fixed point, the normalized vector field would extend continuously across the disc and its boundary map would have degree zero. This contradicts the index $+1$. Thus every planar periodic orbit encloses a fixed point.