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There are two ways to compute Poincar\'e-Dulac normal forms of systems ofODEs. Under the original approach used by Poincar\'e the normalizingtransformation is explicitly computed. On each step, the normalizing procedurerequires the substitution of a polynomial to a series. Under the otherapproach, a normal form is computed using Lie transformations. In this case,the changes of coordinates are performed as actions of certain infinitesimalgenerators. In both cases, on each step the homological equation is solved inthe vector space of polynomial vector fields $V^n_j$ where each component ofthe vector field is a homogeneous polynomial of degree $j$. We present thethird way of computing normal forms of polynomial systems of ODEs where thecoefficients of all terms are parameters. Although we use Lie transforms thehomological equation is solved not in $V^n_j$ but in the vector space ofpolynomial vector fields where each component is a homogeneous polynomial inthe parameters of the system. It is shown that the space of the parameters is akind of dual space and, the computation of normal forms can be performed in thespace of parameters treated as the space of generalized vector fields. Theapproach provides a simple way to parallelize the normal form computationsopening the way to compute normal forms up to higher order than underpreviously known two approaches.

Computation of Normal Forms for Systems with Many Parameters