Amplitude-phase equations for a weakly perturbed oscillator
= Amplitude-phase equations for a weakly perturbed oscillator
= Amplitude-phase equations
{synonym}
For $u''+u=\epsilon f(u,u',\epsilon t)$, write $u_0=R(T)\cos(t+\phi(T))$ with $T=\epsilon t$. Removing resonant sine and cosine forcing gives
$$
R'=-\langle f\sin(t+\phi)\rangle,
\qquad
R\phi'=-\langle f\cos(t+\phi)\rangle.
$$