= Nilpotent cubic damping invariant
{title2=$b=g_{21}+3f_{30}$}
For a reflection-symmetric <vector field> $\dot u=v+f_{30}u^3+f_{21}u^2v+\cdots$, $\dot v=g_{30}u^3+g_{21}u^2v+\cdots$, introduce $x=u$, $y=\dot u$. Differentiating gives $\dot x=y$ and a cubic acceleration with $x^2y$ coefficient $b=g_{21}+3f_{30}$. The contribution $3f_{30}$ comes from differentiating the first equation's cubic term. Further cubic <near-identity transformations> can remove nonresonant terms but cannot turn $b=0$ into a nonzero coefficient by a nonsingular scaling. This is a useful check before assigning a generic reflection-symmetric <Bogdanov–Takens bifurcation> to a calculated <Taylor expansion>.
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