= Solution
A <Lyapunov function> for an equilibrium is a continuously differentiable function that is positive definite there and whose derivative along nonconstant trajectories is negative definite in a neighbourhood.
For $V=x^2+ky^2$, positivity requires $k>0$, and
$$
\dot V
=-2ax^2-2aky^2+2(3-k)xy
+2(x^2+y^2)(x^2+ky^2).
$$
The quartic term is negligible sufficiently close to the origin, so $V$ is a local Lyapunov function exactly when
$$
ax^2+aky^2-(3-k)xy
$$
is positive definite. Its determinant condition is
$$
4a^2k>(3-k)^2.
$$
The two roots of equality are
$$
k_{1,2}(a)=3+2a^2\mp2a\sqrt{a^2+3}
=\left(\sqrt{a^2+3}\mp a\right)^2.
$$
Therefore
$$
\boxed{
(\sqrt{a^2+3}-a)^2<k<(\sqrt{a^2+3}+a)^2
}.
$$
Back to article page
