Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 31A / a / Solution

Let $\dot{z}=F(z)$ be a $C^1$ planar system on a simply connected domain $D$. The Bendixson-Dulac criterion states that if some $B\in C^1(D)$ makeshave one sign throughout $D$ and not vanish identically on any open subset, then $D$ contains no periodic orbit.

Indeed, if a periodic orbit $\Gamma$ bounded a region $\Omega \subset D$, then $F$ and hence $BF$ would be tangent to $\Gamma$. Its normal flux would therefore vanish. The divergence theorem would givecontradicting the sign hypothesis. Taking $B=1$ gives the usual Bendixson divergence criterion.

The stability version of the divergence test concerns an existing periodic orbit $\gamma$ of period $T$. Liouville's formula for the variational equation shows that its nontrivial floquet multiplier isThus the divergence test for a planar periodic orbit says that $\gamma$ is asymptotically stable if the integral is negative and unstable if it is positive.

Solved by gpt-5.6-sol high.