View source for Poincaré-Dulac normal form

Let <math>X</math> be a <math>C^\infty</math> vector field on <math>\mathbb{R}^n</math> with the origin as an equilibrium point; that is, <math>X(0)=0</math>. <math>X(x(t))=x'(t)</math>. Suppose also that
<math>x'=Ax+f(x)</math>, where <math>A</math> is linear and <math>f(0)=0, D_0 f = 0</math>. 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 <math>y'=Ay</math>, like how with the Morse lemma we can actually get rid of higher order terms around a nondegenerate critical point.

We define a vector <math>\lambda=(\lambda_1,\ldots,\lambda_n) \in \mathbb{R}^n</math> to be resonant if there is some <math>r=(r_1,\ldots,r_n) \in \mathbb{Z}^n</math> with all the <math>r_k \geq 0</math> and with at least two of them nonzero. The order of the resonance is the <math>l^1</math> norm of this integer vector: <math>|r|=\sum r_k</math>.

Poincar\'e's theorem on normal forms: If <math>A</math> has <math>n</math> distinct eigenvalues and they are nonresonant, there exists a formal change of variables <math>x=y+O(|y|^2)</math>, transforming the equation <math>x'=Ax+f(x)</math> to <math>y'=Ay</math>. 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.