= Solution
For the unperturbed oscillator take
$$
x=R\cos t,\qquad y=-R\sin t,
$$
and energy $E=(x^2+y^2)/2$. Its change over one nearly circular orbit is
$$
\Delta E=\varepsilon\int_0^{2\pi}(1-x^2+ay^2)y^2dt
=\varepsilon\pi\left(R^2+\frac{3a-1}{4}R^4\right).
$$
A nonzero balance therefore requires
$$
R^2=\frac4{1-3a},
$$
which exists exactly when $a<1/3$. The averaged energy drift is positive below this radius and negative above it, so the resulting <limit> cycle is stable.
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