Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 76 / 2 / i / Solution

For the rotating-frame normalization of resonant amplitude forcing, use a rotating phase, $A=B\exp [i(\tilde{\Omega }T+\delta )]$. The real $\delta$ must occur inside the factor of $i$; the printed change of variables omits that $i$ on the phase constant. The correctly phased substitution givesA negative real forcing coefficient can be made positive by changing the forcing phase. Assuming $c>0$, setAll three terms then have the same coefficient scale, givingRenaming $\mathcal{T}$ as $T$ gives the requested canonical equation. This rescaling preserves forward time. If $c<0$, a forward-time normalization instead leaves a positive cubic term, and if $c=0$ a cubic normalization is impossible. If $e=0$, the unforced Landau amplitude equation must be treated separately. Thus the printed canonical form implicitly concerns saturation in a supercritical bifurcation with nonzero forcing.