Dihedral fourfold Hopf normal form (${\dot{q}}_1=(\mu +i\omega )q_1-a{q}_1|q_1{|}^2-b{q}_1|q_2{|}^2-c{\bar{q}}_1{q}_2^2$)

In Hopf coordinates where the rotation acts as $(i{q}_1,-i{q}_2)$, reflection swaps components, and temporal phase multiplies both equally, cubic equivariance permits precisely the three displayed monomials in the first component. Swapping indices gives the second equation, with the same complex coefficients. The rotating, axial standing, and diagonal standing branches are $q_2=0$, $q_2=q_1$, and $q_2=i{q}_1$. Their squared component amplitudes are $\mu /\mathrm{Re}a$, $\mu /\mathrm{Re}(a+b+c)$, and $\mu /\mathrm{Re}(a+b-c)$, respectively, when positive.

// Target: dynamical-systems.bigb