Past exam of the differential equations and applications course of the University of Oxford / 2008 / A / 1 / c

Use the method of multiple scales with independent fast and slow variables $t$ and $\tau =ϵt$, and write $x_0=R(\tau )\cos \Theta$, $\Theta =3t+\Phi (\tau )$. At the next order, $x_{1,tt}+9x_1=-2x_{0,t\tau }-(4x_0^2-1)x_{0,t}$. Its resonant terms areSuppressing secular terms yieldsThe leading initial data are $\boxed{R_0(0)=4,{\Phi }_0(0)=0}$, up to adding an integer multiple of $2\pi$ to the phase. Thus $x(t)=R_0(ϵt)\cos [3t+{\Phi }_0(ϵt)]+O(ϵ)$. The amplitude decreases from four towards the stable value one, consistent with the selected limit cycle; the zero-amplitude state is unstable. Order-$ϵ$ corrections enforce the exact initial derivative after differentiating the slow amplitude term.