Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 32A / b / iii / Solution

Yes. On $Q\le 1$ the same inequality gives $\dot{Q}\le 0$, so the closed ellipse is positively invariant. Equality in $\dot{Q}=0$ occurs only at the origin and at the boundary pointAt that point the vector field isso $\{p\}$ is not invariant. The largest invariant subset of $\{\dot{Q}=0\}$ is therefore the origin. LaSalle's invariance principle shows that every trajectory in the closed ellipse converges to the origin. The inequality may consequently be extended to $Q\le 1$.

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