= Signed cyclic three-mode amplitude equations
Independent sign changes of three amplitudes and a cyclic coordinate permutation force each cubic component to be its own amplitude times a cyclic permutation of one quadratic polynomial. One convenient parametrization is $\dot x=x(\mu-aX-cy^2+ez^2)$ with cyclic companions and $X=x^2+y^2+z^2$. The axial and diagonal isotropy types give primary branches; two-coordinate branches can emerge in secondary <pitchfork bifurcations> when cross-couplings change sign.
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