= Solution
Use the <method of multiple scales> with independent fast and slow variables $t$ and $\tau=\epsilon t$, and write $x_0=R(\tau)\cos\Theta$, $\Theta=3t+\Phi(\tau)$. At the next order, $x_{1,tt}+9x_1=-2x_{0,t\tau}-(4x_0^2-1)x_{0,t}$. Its resonant terms are
$$
[6R'+3R(R^2-1)]\sin\Theta+6R\Phi'\cos\Theta.
$$
Suppressing <secular terms> yields
$$
\boxed{\frac{dR_0}{d\tau}=\frac12R_0(1-R_0^2),\qquad\frac{d\Phi_0}{d\tau}=0.}
$$
The leading initial data are $\boxed{R_0(0)=4,\ \Phi_0(0)=0}$, up to adding an integer multiple of $2\pi$ to the <phase>. Thus $x(t)=R_0(\epsilon t)\cos[3t+\Phi_0(\epsilon t)]+O(\epsilon)$. The <amplitude> decreases from four towards the stable value one, consistent with the selected <limit cycle>; the zero-<amplitude> state is unstable. Order-$\epsilon$ corrections enforce the exact initial <derivative> after differentiating the slow <amplitude> term.
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