Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 74 / 2 / a / ii

For velocity-only forcing in averaged oscillator equations, $f=f(\dot{y})$, the oscillation phase average vanishes: $f(-R\sin \varphi )\cos \varphi$ is a periodic total derivative, using an antiderivative of $f$. Hence ${\theta }_T=0$ to first order, although the amplitude may change.

In the cubic case, $f=-R^3{\sin }^3\varphi$, soWith $R(0)=R_0>0$, integration givesFor positive $\epsilon$ this is cubic velocity anti-damping, not damping: the exact energy derivative is $d({\dot{y}}^2+y^2)/(2dt)=\epsilon {\dot{y}}^4\ge 0$. The slow solution grows and formally diverges at $T=4/(3R_0^2)$. The weakly nonlinear approximation loses validity before its amplitude becomes so large that $\epsilon R^2$ is not small; its divergence is not a controlled prediction of the exact late-time solution. If $R_0=0$, the exact zero solution should be used separately.