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In a Hopf critical eigenspace, equip the real space $V\oplus V$ with the complex structure defined by the temporal phase action. A subgroup $\Sigma \subseteq G\times S^1$ is a complex axial isotropy subgroup when it is the full stabilizer of a nonzero vector andThe isotropy is for the combined spatial and temporal action, so its elements encode spatiotemporal symmetry of a periodic orbit, not necessarily pointwise spatial symmetry. “Complex dimension” here refers to the Hopf complex structure; it need not agree with independently chosen spatial complex coordinates. Any kernel of the combined action belongs to every actual isotropy subgroup.

Let a smooth $G$-equivariant vector field have a trivial equilibrium branch and a critical real eigenspace $V\oplus V$, where $V$ is an absolutely irreducible real representation of the finite group $G$. Assume the critical linearization is semisimple with eigenvalues $\pm i{\omega }_0$, ${\omega }_0\ne 0$, no other imaginary-axis eigenvalues are present, and the real part of the critical eigenvalue crosses zero transversely as the parameter varies. The equivariant Hopf theorem then gives, generically, a branch of small-amplitude periodic solutions for each conjugacy class of complex axial isotropy subgroup of $G\times S^1$ in the critical representation. The restricted Hopf bifurcation on its two-dimensional fixed space must be nondegenerate, with nonzero leading radial cubic coefficient. Each branch has the associated spatiotemporal symmetry of a periodic orbit and period tending to $2\pi /|{\omega }_0|$.

The spatial multiplicity means this is not the ordinary simple-eigenvalue Hopf hypothesis: the required critical representation is the whole two-copy isotypic component, and the semisimple pair has the corresponding symmetry multiplicity. The theorem guarantees primary branch existence and symmetry, but neither their stability nor the side of the parameter on which they exist. Those require the nonlinear coefficients and transverse perturbation analysis. A centre manifold and a symmetry-compatible normal form of a dynamical system isolate the critical dynamics and supply those coefficients.

The proposed transformation diagonalizes the time-circle action, but the two resulting coordinates have opposite temporal weights. Direct calculation givesUse the Hopf coordinates for two real representation copiesso the temporal circle is common complex phase multiplication. Writing its phase action as $T_{\theta }q=e^{i\theta }q$, with $\theta =-\varphi$, the spatial generators become $\rho (q_1,q_2)=(i{q}_1,-i{q}_2)$ and $m_x(q_1,q_2)=(q_2,q_1)$. The combined action has kernel $K=\{(1,1),({\rho }^2,-1)\}$.

Three nonconjugate complex axial isotropy subgroups and their fixed complex lines areFor example $(\rho ,e^{-i\pi /2})$ acts as $(q_1,-q_2)$, forcing $q_2=0$; reflection forces $q_2=q_1$; and $(\rho m_x,-1)$ acts as $(-i{q}_2,i{q}_1)$, forcing $q_2=i{q}_1$. Each is the full stabilizer of a generic nonzero point of that line and includes $K$. The rotational spatial projection of ${\Sigma }_R$ distinguishes it from the standing types; the other two use the two nonconjugate reflection-axis classes in $D_4$.

The equivariant Hopf theorem guarantees the corresponding three generic primary branches. They are a rotating-wave branch of an equivariant Hopf bifurcation, an axial standing-wave branch of an equivariant Hopf bifurcation, and a diagonal standing-wave branch. In the literal $z$ coordinates their fixed spaces are $z_2=0$, $z_2={\bar{z}}_1$, and $z_2=-i{\bar{z}}_1$, respectively; using a common $e^{i\Omega t}$ in both original $z$ coordinates would not describe these standing branches correctly.

To interpret them physically, invert the transformation: $w_1=z_1+z_2$, $w_2=-i(z_1-z_2)$. For $q=R{e}^{i\Omega t}$, the three representatives areThe rotating solution changes spatial direction throughout the cycle. The standing solutions remain on a fixed spatial axis or diagonal while the two real-copy amplitudes oscillate in quadrature. Reflections exchange the two rotating chiralities; rotations generate the other representatives within each standing-wave type.

First there is a necessary coordinate qualification. With the literal $z$ action obtained in part (c), the temporal weights are $-1,+1$ and reflection is conjugation followed by exchange. A linear term with $(\mu +i\omega )$ in both equations fails reflection equivariance for $\omega \ne 0$, since conjugation changes the sign of $i\omega$. The proposed first-component monomial ${\bar{z}}_1{z}_2^2$ also has temporal weight $+3$, rather than $-1$. Thus the displayed equations cannot be proved in those literal coordinates. They are the correct dihedral fourfold Hopf normal form in the common-phase Hopf coordinates $(q_1,q_2)=(z_1,{\bar{z}}_2)$.

In these coordinates the temporal circle imposes total phase weight one on every vector-field monomial. There are no quadratic monomials with that weight. A cubic monomial has two unbarred factors and one barred factor. For the first component, rotation by $\rho$ must multiply the monomial by $i$. Inspecting the six phase-compatible monomials leaves preciselyThe excluded three monomials multiply by $-i$ instead. Reflection swaps the components, so their coefficients must be the same after swapping indices. The linear part is a scalar complex multiple by absolute irreducibility. Orient the Hopf parameter to write it as $\mu +i\omega$, and name the cubic coefficients $-a,-b,-c$, where $a,b,c$ are generally complex. This provesThese are cubic amplitude normal forms of a dynamical system, after the usual symmetry-compatible Hopf normalization. Returning to literal $z$ coordinates conjugates the second equation: its linear coefficient is $\mu -i\omega$, its cubic coefficients are $\bar{a},\bar{b},\bar{c}$, and the first coupling is $-c{\bar{z}}_1{\bar{z}}_2^2$. That corrected form respects every printed generator.

On the three complex axial isotropy subgroup fixed spaces, substitute $q_1=R{e}^{i\Omega t}$ and $q_2=0,q_1,i{q}_1$. The radial equations are $\dot{R}=R(\mu -A_j{R}^2)$, withTherefore the leading component amplitudes areEach expression applies on the parameter side where it is positive, with its denominator nonzero for a nondegenerate branch. The frequencies are ${\Omega }_j=\omega -\mathrm{Im}(a_j)R_j^2$, with $a_j=a,a+b+c,a+b-c$. The denominators are generically distinct, so coexisting branches generically have different component amplitudes. If the amplitude is instead measured by the original real-space norm, the identity $|w_1{|}^2+|w_2{|}^2=2(|q_1{|}^2+|q_2{|}^2)$ gives squared amplitudes $2R_R^2$, $4R_A^2$, and $4R_D^2$, again generically distinct. Equalities require special coefficient relations, rather than symmetry forcing them.