= Cubic elimination at a nilpotent double-zero point
{title2=$P=a_2,\quad Q=b_2+3a_1$}
For a <vector field> with linear part $\dot u=v$, $\dot v=0$ and cubic terms $a_i u^3+b_i u^2v+c_iuv^2+d_iv^3$, write $(u,v)=(x,y)+(h_1,h_2)$ with homogeneous cubic $h_i$. The transformed cubic field is $f_3+Ah-Dh\,A(x,y)^T$, where $A(x,y)^T=(y,0)^T$. Taking the coefficients of $h_2$ to be $(-a_1,c_2/2,d_2,0)$ and those of $h_1$ to be $((b_1+c_2/2)/3,(c_1+d_2)/2,d_1,0)$ eliminates every cubic term in $\dot x$ and the $xy^2,y^3$ terms in $\dot y$. The remaining <normal form> is $\dot x=y$, $\dot y=Px^3+Qx^2y$ through cubic order. The invariant damping coefficient contains the contribution $3a_1$ from differentiating the first equation.
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