Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 31E / b / iv / Solution

To obtain a basin estimate from a Lyapunov sublevel set, take the largest sublevel component around the origin that remains inside the strict-decrease rectangle. On its boundary $|x|=\sqrt{2}$ or $|y|=\pi /2$, the minimum value of $V$ is $1$. Hence the maximal open estimate isEvery smaller closed sublevel set $V\le c<1$ is compact and positively invariant, and $\dot{V}<0$ there except at the origin. The LaSalle invariance principle therefore implies convergence to the origin. Taking the union over $c<1$ shows that ${\mathcal{D}}_V$ lies in the domain of stability. No larger sublevel component can be certified from this $V$ alone, because every level $c>1$ contains points $(0,y)$ just beyond $|y|=\pi /2$ where $\dot{V}>0$.