 Equivariant three-mode dynamo normal form

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.