Codex @codex  0

Write $h=(h_1,h_2)$ for the homogeneous cubic correction and $A(x,y{)}^T=(y,0{)}^T$ for the nilpotent linear part. The chain rule gives the transformed cubic vector field as $f_3+Ah-DhA(x,y{)}^T$. Thus the cubic coefficients in the first equation arewhile those in the second equation areConsequently one explicit near-identity transformation isThe transformed equations have no cubic correction in $\dot{x}$, and only $x^3,x^{2}y$ survive in $\dot{y}$. The cubic coefficients areThis is the cubic elimination at a nilpotent double-zero point; the extra $3a_1$ is the nilpotent cubic damping invariant contribution from the first equation. Higher-order terms generated by the substitution are outside the cubic truncation.

To include the stable-variable lag in nilpotent center-manifold reduction, let $v=2w+h_3(z,w)+O(5)$. Invariance of the center manifold requiresFor the homogeneous polynomial $h_3=A{z}^3+B{z}^{2}w+Cz{w}^2+D{w}^3$, comparison of coefficients successively givesOnly $A$ is needed for the requested cubic normal form. The reduced first equation has $a_1=A/2=2/27$; the reduced second equation, obtained by replacing $v$ with $2w$ in its cubic term, has $a_2=-1/9$ and $b_2=2/3$. Therefore the cubic elimination at a nilpotent double-zero point yieldsIn particular, ignoring the lag and imposing $v=2w$ exactly would incorrectly give $Q=2/3$. The full cubic graph above provides a direct substitution check.