 Stable-variable lag in nilpotent center-manifold reduction

For $\dot{z}=v/2$, $\dot{v}=-3v/2+3w$, $\dot{w}=-(z-v{)}^3/9$, the stable coordinate is $v-2w$ and the center manifold begins with $v=2w$. A cubic correction $h_3$ satisfies $w{\partial }_z{h}_3-(2/9)(z-2w{)}^3=-3h_3/2$, giving $h_3=4z^3/27-32z^{2}w/27+272z{w}^2/81-832w^3/243$. The derivative of the graph contributes at the same order as the cubic forcing. Simply setting $\dot{v}=0$ omits this lag and incorrectly changes the cubic damping coefficient of the reduced normal form.