= Rotating-frame normalization of resonant amplitude forcing
{title2=$C_T+i\omega C=\mu C-|C|^2C+\overline C^{\,2}$}
For the <three-to-one spatially forced amplitude equation> with forcing $e\exp[3i(\Omega T+\delta)]$, put $A=B\exp[i(\Omega T+\delta)]$. The real phase constant $\delta$ belongs inside the factor of $i$. If $c>0$ and the phase has been chosen so that $e>0$, the scaling $B=(e/c)C$, $\mathcal T=(e^2/c)T$ gives the displayed canonical <amplitude equation>, with $\mu=c\widetilde\mu/e^2$ and $\omega=c\Omega/e^2$. A negative cubic saturation coefficient cannot give the same cubic sign under a forward-time normalization; zero forcing or zero cubic coefficient requires another scaling.
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