= Threefold phase-locked equilibria
{title2=$R_\pm^2=\mu+\frac12\pm\frac12\sqrt{1+4\mu-4\omega^2}$}
The canonical <three-to-one spatially forced amplitude equation> has nonzero <equilibrium points> $C=Re^{i\theta}$ satisfying $\cos3\theta=(R^2-\mu)/R$ and $\sin3\theta=-\omega/R$. Thus $(R^2-\mu)^2+\omega^2=R^2$. Each positive amplitude has three phases separated by $2\pi/3$: two amplitude branches normally mean six complex <equilibrium points>. For $\mu>\omega^2-1/4$ the larger branch is asymptotically stable and the smaller consists of <saddle equilibria>, except that its zero-amplitude root at $(\mu,\omega)=(0,0)$ is not a nonzero <equilibrium point>. Equality gives a <saddle-node bifurcation>. This <phase locking> breaks continuous translation symmetry down to threefold symmetry.
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