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 $\dot{x}=x(\mu -aX-c{y}^2+e{z}^2)$ 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.

When $c=e$ in the signed cyclic three-mode amplitude equations, $\dot{X}=2X(\mu -aX)$ and $\dot{V}=3V(\mu -aX)$ for $V=xyz$. Thus $V/X^{3/2}$ is conserved wherever $X>0$, precluding asymptotically stable isolated equilibria. On $X=\mu /a$, each octant contains closed contours of $V$ around the equal-magnitude equilibrium, bounded by heteroclinic arcs between coordinate axes if $c\ne 0$. For $c=0$ the sphere consists entirely of equilibria.