Rational normal form for dynamical systems by Carleman linearization

Abstract

We propose in this paper a rational normal form for dynamical systems or vector elds. That is to say if the coecients of the system are in a eld K (which, in practice, is Q;R), so is the computed rational normal form. We give an algorithm for an eective computation of the normal form up to a nite order. We don't need to compute the Jordan canonical form of the leading matrix neither its eigenvalues. We give also a rational method to test the existence of resonance of a xed order without computing the eigenvalues of the linear part. Our normal form is a renement of the classical normal forms. Our method is applicable for both the nilpotent and the non nilpotent cases.