= Phase locking of a resonant hexagonal triad
{title2=$\dot\Phi=-\alpha\left(\frac{r_2r_3}{r_1}+\frac{r_3r_1}{r_2}+\frac{r_1r_2}{r_3}\right)\sin\Phi$}
For $A_i=r_ie^{i\phi_i}$ with all $r_i>0$, the translation-invariant total phase is $\Phi=\phi_1+\phi_2+\phi_3$. With $\alpha>0$, the displayed equation makes $\Phi=0$ stable and $\Phi=\pi$ unstable. A spatial translation can remove two phases, so the stable locked states have all amplitudes positive real in a suitable origin. This is a statement modulo translation, not that every translated representative is real; the zero solution and rolls belong to boundary strata.
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