= Solution
Apply the <method of multiple scales> with independent fast time $t$ and <slow time> $T=\varepsilon t$, and write $x=x_0(t,T)+\varepsilon x_1(t,T)+\cdots$. The chain rule gives $d/dt=\partial_t+\varepsilon\partial_T$. At leading order,
$$
(\partial_t^2+1)x_0=0,\qquad x_0=R(T)\cos\theta,\qquad \theta=t+\phi(T).
$$
At first order,
$$
(\partial_t^2+1)x_1=f(R\cos\theta,-R\sin\theta)+2R_T\sin\theta+2R\phi_T\cos\theta.
$$
A forcing component at the natural <frequency> produces a <secular term> in $x_1$. Thus the <solvability condition in the method of multiple scales> is orthogonality to both $\sin\theta$ and $\cos\theta$. Define the fast-phase average, holding $R,\phi,T$ fixed, by
$$
\langle h\rangle=\frac1{2\pi}\int_0^{2\pi}h(\theta;R,T)\,d\theta.
$$
Since $\langle\sin^2\theta\rangle=\langle\cos^2\theta\rangle=1/2$ and $\langle\sin\theta\cos\theta\rangle=0$, the <amplitude-phase equations> are
$$
\boxed{R_T=-\langle f\sin\theta\rangle,\qquad R\phi_T=-\langle f\cos\theta\rangle.}
$$
They are leading averaged equations, not exact equations for a purely sinusoidal solution of a general nonlinear oscillator.
For the cubic conservative force, $f=-R^3\cos^3\theta$, odd symmetry gives $\langle\cos^3\theta\sin\theta\rangle=0$, while $\langle\cos^4\theta\rangle=3/8$. Hence
$$
\boxed{R(T)=R_0,\qquad\phi(T)=\phi_0+\frac38R_0^2T.}
$$
If $x(0)=x_*$ and $x'(0)=v_*$, leading initial matching gives
$$
R_0=\sqrt{x_*^2+v_*^2},\qquad \phi_0=\operatorname{atan2}(-v_*,x_*),
$$
and therefore
$$
x(t)=R_0\cos\!\left(t+\phi_0+\frac38\varepsilon R_0^2t\right)+O(\varepsilon)
$$
on bounded <slow time> intervals for fixed bounded initial data. The positive <frequency> shift agrees with the increased restoring force of $x''+x+\varepsilon x^3=0$. The zero initial state is the exact zero solution, with irrelevant phase. If a first waveform correction is desired, the remaining first-order forcing is $-R_0^3\cos3\theta/4$, so $x_1=R_0^3\cos3\theta/32$ plus a homogeneous correction chosen to match the initial data at the next order. This exhibits explicitly why the leading harmonic ansatz is approximate.
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