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For a linear group action of $G$ on a state space, an equivariant dynamical system has a vector field satisfyingThus applying a symmetry before computing a velocity gives the same result as applying it to the computed velocity. An equivariant vector field need not be invariant as a vector: its value transforms along with the point where it is evaluated.

If the ordinary differential equation has unique solutions, its flow map obeys ${\Phi }_t(gx)=g{\Phi }_t(x)$ whenever both sides exist. Indeed, $g{\Phi }_t(x)$ has derivative $gF({\Phi }_t(x),\mu )=F(g{\Phi }_t(x),\mu )$ and starts at $gx$, so Picard-Lindelöf theorem identifies it with ${\Phi }_t(gx)$.

Consequently, every subgroup's fixed-point subspace of a group action is invariant under the flow map. If $hx=x$ for all $h$ in the subgroup, then $h{\Phi }_t(x)={\Phi }_t(hx)={\Phi }_t(x)$. This reduces some symmetry-preserving motions to a smaller state space. It also constrains normal forms: the wavevector selection rule for equivariant monomials and the cubic equivariants of the full cube symmetry group are concrete coefficient tests.

Coordinate reflection in $x_i$ requires the $i$th component to be odd in $x_i$, and reflections in the other coordinates require it to be even in those coordinates. Thus no constant or quadratic terms are permitted. The only cubic monomials in component $i$ are $x_i^3$ and $x_i{x}_j^2$ for $j\ne i$. Coordinate permutations equate their coefficients in all components. After using the transverse linear eigenvalue as parameter, the normal form isThe remainder starts at fifth spatial order, together with parameter-dependent corrections such as $\mu |x{|}^3$. This is the cubic equivariants of the full cube symmetry group form. On a branch with $m=1,2,3$ equal nonzero absolute coordinates $s$, respectively,Hence the three leading branch amplitudes are $s^2=-\mu /a$, $-\mu /(a+b)$ and $-\mu /(a+2b)$. Their generic nonzero denominators determine on which side the branches occur; existence alone does not make them stable.