= Cubic equivariants of the full cube symmetry group
{title2=$\dot x_i=\mu x_i+a x_i^3+b x_i\sum_{j\ne i}x_j^2+\cdots$}
For the natural action of all signed coordinate permutations on $\mathbb R^3$, a cubic <equivariant dynamical system> has components
$$
\dot x_i=\mu x_i+a x_i^3+b x_i\sum_{j\ne i}x_j^2+\cdots.
$$
Reflecting $x_i$ alone reverses component $i$, so that component must be an <odd function> of $x_i$. Reflecting any other coordinate leaves component $i$ fixed, so it must be an <even function> of each other coordinate. Consequently the only cubic <monomials> are $x_i^3$ and $x_ix_j^2$. Permuting the other coordinates makes their coefficients equal; permuting all coordinates gives the same pair $a,b$ in every component. The same parity test excludes every quadratic term.
A branch with exactly $m\in\{1,2,3\}$ nonzero coordinates of common magnitude $r$ satisfies
$$
0=r\bigl(\mu+[a+(m-1)b]r^2\bigr).
$$
Thus its nonzero states require $r^2=-\mu/[a+(m-1)b]>0$, provided the denominator is nonzero. The three cases describe coordinate-axis, face-diagonal and body-diagonal states. If the denominator vanishes, the cubic truncation cannot determine that branch's amplitude; higher-order terms must be retained. Existence of a branch does not by itself determine its <stability>.
Back to article page
