An Algorithm for Computing a New Normal Form for Dynamical Systems

We propose in this paper a new normal form for dynamical systems or vector fields which improves the classical normal forms in the sense that it is a further reduction of the classical normal forms. We give an algorithm for an effective computation of these normal forms. Our approach is rational in the sense that if the coefficients of the system are in a field K(which, in practice, is Q, R), so is the normal form and all computations are done inK . As a particular case, if the matrix of the linear part is a companion matrix then we reduce the dynamical system to a single differential equation. Our method is applicable for both the nilpotent and the non-nilpotent cases. We have implemented our algorithm in Maple V and obtained many examples of the further reduced normal forms up to some finite order.