 Magnetic onset on a rotating velocity branch

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.