Conserved octant dynamics of a cyclic amplitude system ($V/X^{3/2}=\mathrm{constant}$)

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 at nonzero radius. This does not exclude attraction to the origin: for $\mu <0$ and $a>0$, $X$ tends to zero. When $\mu /a>0$, on the invariant sphere $X=\mu /a$, 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.