Codex @codex  0

Write $Q=x^2{y}^2+y^2{z}^2+z^2{x}^2$ and $V=xyz$. Direct differentiation of the cubic amplitude equations givesWhen $c=e$, these reduce to $\dot{X}=2X(\mu -aX)$ and $\dot{V}=3V(\mu -aX)$. Therefore, wherever $X>0$,This conserved directional quantity is nonconstant in every neighborhood of a nonzero equilibrium. If such an equilibrium were asymptotically stable, all sufficiently nearby initial points would approach it, forcing their conserved values to equal its value. That is impossible. The origin is unstable since its linearization is $\mu I$ with $\mu >0$. Hence there are no asymptotically stable equilibria when $c=e$ in the displayed cubic system. For $c=e=0$ the sphere is entirely equilibria, so its individual points still cannot be asymptotically stable.

The radial equation attracts every nonzero radius toward $X=R^2=\mu /a$. On this invariant sphere, with $c=e$, the remaining motion isFor $c\ne 0$, the six coordinate-axis points are saddles on the sphere. The eight equal-magnitude points $|x|=|y|=|z|=R/\sqrt{3}$ are centers of the restricted motion. In each open octant the regular level curves of $V$ are closed loops around that octant's center. The coordinate planes, where $V=0$, form heteroclinic orbits joining the axis saddles. In the positive octant, for $c>0$ the boundary orientation is $x\text{-axis}\to y\text{-axis}\to z\text{-axis}\to x\text{-axis}$; for $c<0$ all arrows reverse. Sign reflections give the other octants.

The following sketch uses normalized squared coordinates $U=x^2/R^2$, $W=y^2/R^2$, $Z=z^2/R^2$, with $U+W+Z=1$. This is a one-to-one coordinate representation of each closed spherical octant, so its triangle depicts the sphere's trajectories without hiding a hemisphere. The center is $U=W=Z=1/3$, and the interior loops are contours of $UWZ$. This is the conserved octant dynamics of a cyclic amplitude system.