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  "body": "\n## Definition\n\nLet $A$ be a real $n \\times n$ matrix and consider the [[autonomous system|07BNRFQ0PJ]]\n$$x' = A x + f_2(x) + f_3(x) + \\cdots + f_r(x) + O(\\abs{x}^{r+1})$$\nnear the [[equilibrium point|0G4X36AE0X]] $0$ of $\\R^n$.\nHere each $f_m$ is a vector field whose components are [[homogeneous polynomials|2C3PAQ7H50]] of degree $m$.\nLet $H_m$ be the vector space of such vector fields of degree $m$.\nThe homological operator $L_A \\colon H_m \\to H_m$ is $L_A h(x) = Dh(x) \\, A x - A h(x)$.\nFor each $m \\ge 2$ choose a complement $G_m$ of the image $L_A(H_m)$ in $H_m$.\nThe system is in normal form up to order $r$ with respect to these complements if $f_m \\in G_m$ for $m = 2, \\ldots, r$.\n\n## Remarks\n\nThe 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 $r$.\nAt 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 $G_m$.\nIf $A = \\mathrm{diag}(\\lambda_1, \\ldots, \\lambda_n)$, 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.\nSo 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.\nExample: 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 $c$.\nNon-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.\n\n## Explanations\n\n### L1 human\n\nA normal form is the result of simplifying the nonlinear terms of an equation by a change of variables, one degree at a time.\nThe terms that survive are the resonant ones, which reflect relations between the [[eigenvalues|0BQAA2BK3Q]] of the linear part.\n",
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