Equivariant dynamical system ($F(gx,\mu )=gF(x,\mu )$)

An equivariant dynamical system has a vector field commuting with the action of a symmetry group. Its flow map preserves every subgroup's fixed-point subspace, because equivariance and uniqueness keep initial states fixed by that subgroup. Polynomial normal forms are constrained by the weights and parity of the group representation.