= Chiral hexagon Hopf threshold
{title2=$r_H=2\alpha/(b+c-2a)$}
For equal positive amplitudes $r$, let $s=a+b+c$ and use $\mu+\alpha r-sr^2=0$. The radial Jacobian eigenvalue is $\alpha r-2sr^2$. The conjugate pair has real part $(b+c-2a)r^2-2\alpha r$ and imaginary parts $\pm\sqrt3(b-c)r^2$. When $b+c>2a>0$ and $b\ne c$, the upper hexagon branch loses amplitude stability through this pair at the displayed radius. The associated parameter is $\mu_H=2\alpha^2(4a+b+c)/(b+c-2a)^2$. Nonlinear terms decide the direction and stability of the resulting periodic branch.
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