= Conserved octant dynamics of a cyclic amplitude system
{title2=$V/X^{3/2}=\mathrm{constant}$}
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. On $X=\mu/a$, each octant contains closed contours of $V$ around the equal-magnitude equilibrium, bounded by heteroclinic arcs between coordinate axes if $c\ne0$. For $c=0$ the sphere consists entirely of equilibria.
Back to article page
