Equivariant three-mode dynamo normal form ($B\in \mathbb{R},A,V\in \mathbb{C}$)

An axisymmetric magnetic mode is unchanged by axial rotations, while an equatorial magnetic mode and a corresponding velocity mode transform as $A,V\mapsto e^{i\theta }(A,V)$. Magnetic reversal sends $(B,A,V)$ to $(-B,-A,V)$. For magnetic evolution linear in magnetic amplitudes, rotational equivariance requires $\dot{B}=Bf(|V{|}^2)+2\mathrm{Re}[A^{\ast }Vg(|V{|}^2)]$ and $\dot{A}=Ah(|V{|}^2)+BVj(|V{|}^2)+A^{\ast }{V}^2\ell (|V{|}^2)$, with $f$ real and the other functions complex. Velocity monomials $B^p{A}^q(A^{\ast }{)}^r{V}^s(V^{\ast }{)}^t$ must have even $p+q+r$ and weight $q-r+s-t=1$. These conditions enumerate the cubic normal form without assuming reflection symmetry or nonlinear magnetic self-interactions.

For real $c_2,d_4$ and the simplified magnetic amplitude equations, the pure-velocity branch is $V=s{e}^{i{\omega }_{3}t}$ with $A=B=0$. In its rotating frame the magnetic linearization on $(B,\mathrm{Re}A,\mathrm{Im}A)$ has rows $({\mu }_1-e{s}^2,2s,0)$, $(c_{2}s,{\mu }_2,-\Delta )$ and $(0,\Delta ,{\mu }_2)$, where $\Delta ={\omega }_2-{\omega }_3$. Its zero determinant gives the displayed onset relation. A nonzero magnetic branch needs a kernel involving both magnetic amplitudes, a transverse parameter crossing and nondegenerate nonlinear saturation; determinant zero alone is insufficient in degenerate cases.

For nonzero coefficients of $A^{\ast }V$ in $\dot{B}$ and $AB$ in $\dot{V}$, a relative phase rotation makes $c_1$ real positive. Positive rescalings of $A,V,B$ then normalize both magnitudes to one. The phase of $c_1{c}_3$ is invariant under these rescalings and remains as the phase of normalized $c_3$. Zero couplings cannot be normalized to one; they are degenerate parameter cases.