Second harmonic and mean displacement of a weakly cubic oscillator ($x=A\cos \vartheta -3\lambda A^2/(2m{\omega }^2)+[\lambda A^2/(2m{\omega }^2)]\cos 2\vartheta$)

With $\vartheta =\omega t+\delta$, a weak cubic potential produces a nonzero mean position and a second harmonic at first order. There is no first-order frequency shift. Higher-order frequency changes can accumulate phase over sufficiently long times, so the first-order expression is not a uniform all-time approximation.