Normal form

Remarks

The normal form theorem states that a near-identity polynomial change of coordinates $x = y + h_2(y) + \cdots + h_r(y)$, with $h_m \in H_m$, brings any such system into normal form up to order $r$. At degree $m$ the change $x = y + h_m(y)$ replaces $f_m$ by $f_m - L_A h_m$ and changes only terms of degree at least $m$, so $f_m$ can be reduced to its component in $G_m$. If $A = \mathrm{diag}(\lambda_1, \ldots, \lambda_n)$, then $L_A$ maps the monomial vector field $x_1^{k_1} \cdots x_n^{k_n} e_j$ to $(k_1 \lambda_1 + \cdots + k_n \lambda_n - \lambda_j)$ times itself. So one can take $G_m$ spanned by the resonant monomials, those with $k_1 \lambda_1 + \cdots + k_n \lambda_n = \lambda_j$, and the normal form keeps only resonant terms. Example: for $\lambda = (1, 2)$, the only resonant monomial of degree $2$ is $x_1^2 e_2$, since $2 \cdot 1 = 2$; the normal form up to order $2$ is $y_1' = y_1$, $y_2' = 2 y_2 + c\, y_1^2$ for a constant $c$. Non-example: the system $x_1' = x_1 + x_1 x_2$, $x_2' = 2 x_2$ is not in normal form for $\lambda = (1, 2)$, since $x_1 x_2 e_1$ has $1 + 2 \ne 1$ and is not resonant.