Dihedral threefold Hopf normal form

For the $D_3$ action $\rho (z_1,z_2)=({\zeta }^{-1}{z}_1,\zeta z_2)$, $m(z_1,z_2)=(z_2,z_1)$ and common phase multiplication, the fifth-order normal form has first component $z_1[\alpha +a|z_1{|}^2+b|z_2{|}^2+c|z_1{|}^4+d|z_1{|}^2|z_2{|}^2+e|z_2{|}^4]+f{\bar{z}}_1^2{z}_2^3$ and second component obtained by swapping indices. Here all coefficients may be complex. Its three primary symmetry types are a one-component rotating wave and the two standing-wave lines $z_1=z_2$ and $z_1=-z_2$, together with their group conjugates. The two standing amplitudes split at fifth order through $\mathrm{Re}f$.

If only reflection and common temporal phase remain, the linear complex matrix is $\left(\begin{array}{cc}\alpha & \eta \\ \eta & \alpha \end{array}\right)$. The symmetric and antisymmetric modes have eigenvalues $\alpha +\eta$ and $\alpha -\eta$, generically separating their Hopf thresholds. One-component rotating-wave axes are no longer invariant. Their hyperbolic finite-amplitude periodic solutions can persist as mixed-mode solutions, often through secondary symmetry-breaking bifurcations, while the double primary onset splits into two simple Hopf onsets. Coefficients determine the detailed secondary diagram and stability.