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

The region $D$ is compact and positively invariant and contains the unique fixed point. If that fixed point is unstable, choose an initial point in $D$ which is not on its stable set. Its nonempty compact $\omega$-limit set cannot be the fixed point. The Poincare-Bendixson theorem therefore supplies a periodic orbit. Hence a sufficient parameter condition, and the one selected by the preceding analysis, is