= Solution
Choose \b[$A=B=\sigma/r$] in the proposed <Lyapunov function>. The cubic terms cancel and direct differentiation gives
$$
\dot V=-\sigma x^2+2\sigma xy-\frac\sigma r y^2-\frac{\sigma b}r z^2
=-\sigma(x-y)^2-\sigma(1/r-1)y^2-\frac{\sigma b}r z^2.
$$
For $0<r<1$ this is negative definite, while $V$ is positive definite and radially unbounded. Finite-dimensional equivalence of these positive quadratic forms gives $\dot V\leq-cV$ for some $c>0$. Thus $V(t)\leq V(0)e^{-ct}$, every solution remains bounded and exists for all forward times, and every solution tends to zero. Therefore \b[the origin is globally asymptotically stable], indeed exponentially stable.
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