Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 33A / a / Solution

On each side of the stated parallelogram, substitution into the outward normal component ofshows that the leading $L^3$ terms point inward; the remaining terms are $O(L)$, so the region is positively invariant for sufficiently large $L$.

There is one equilibrium,Its Jacobian has determinant one and trace $\mu (1-F^2)$. If $F^2<1$ the equilibrium is unstable. A trajectory starting nearby remains in the compact trapping region, and its omega-limit set contains no stable equilibrium. The Poincare-Bendixson theorem therefore supplies a periodic orbit.

Solved by gpt-5.6-sol high.