Poincaré-Dulac normal form

Let $X$ be a $C^{\infty }$ vector field on $\mathbb {R} ^{n}$ with the origin as an equilibrium point; that is, $X(0)=0$. $X(x(t))=x'(t)$. Suppose also that $x'=Ax+f(x)$, where $A$ is linear and $f(0)=0,D_{0}f=0$. We would like to make our system as linear as possible, and ideally we would be able to do some change of coordinates to get $y'=Ay$, like how with the Morse lemma we can actually get rid of higher order terms around a nondegenerate critical point.

We define a vector $\lambda =(\lambda _{1},\ldots ,\lambda _{n})\in \mathbb {R} ^{n}$ to be resonant if there is some $r=(r_{1},\ldots ,r_{n})\in \mathbb {Z} ^{n}$ with all the $r_{k}\geq 0$ and with at least two of them nonzero. The order of the resonance is the $l^{1}$ norm of this integer vector: $|r|=\sum r_{k}$.

Poincar\'e's theorem on normal forms: If $A$ has $n$ distinct eigenvalues and they are nonresonant, there exists a formal change of variables $x=y+O(|y|^{2})$, transforming the equation $x'=Ax+f(x)$ to $y'=Ay$. The proof is given in Broer09.

The name Dulac is attached to the case when A has resonant eigenvalues. Then we can no longer push the nonlinearities off to infinity, but we can push most of them away, leaving only so-called resonant monomials.