= Solution
The factor $x^2(2-x^2)$ is positive for $0<|x|<\sqrt2$. The product $y^3\sin2y$ is positive for $0<|y|<\pi/2$. Therefore $\dot V<0$ away from the origin throughout
$$
\boxed{|x|<x_0=\sqrt2,
\qquad |y|<y_0=\frac\pi2}.
$$
Neither side length can be enlarged while retaining a strict Lyapunov function on the whole rectangle: at $(x,0)$ with $|x|>\sqrt2$, the first term makes $\dot V>0$, while at $(0,y)$ immediately beyond $|y|=\pi/2$, the second term makes $\dot V>0$.
Strict decrease on this rectangle does not by itself put the whole rectangle in the <domain of stability>, because the rectangle has not been shown to be <positively invariant set>[positively invariant]; a trajectory may leave it before convergence.
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