Chiral hexagon Hopf threshold ($r_H=2\alpha /(b+c-2a)$)

Assume $\alpha >0$. For equal positive amplitudes $r$, let $s=a+b+c$ and use $\mu +\alpha r-s{r}^2=0$. The radial Jacobian eigenvalue is $\alpha r-2s{r}^2$. The conjugate pair has real part $(b+c-2a)r^2-2\alpha r$ and imaginary parts $\pm \sqrt{3}(b-c)r^2$. 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.