First-order canonical removal of a cubic oscillator perturbation ($x=X-\lambda X^2/(m{\omega }^2)-2\lambda P^2/(m^3{\omega }^4)$)

The accompanying momentum map is $p=P+2\lambda XP/(m{\omega }^2)$. Its Poisson bracket is one up to second-order corrections. Substitution into $p^2/(2m)+m{\omega }^2{x}^2/2+\lambda x^3$ cancels all terms linear in $\lambda$, leaving a harmonic Hamiltonian to that order. The map is local and perturbative, not an exact global removal of an unbounded cubic potential.

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.