Extension of Bautin theory to any dimension

International audienceBautin made some years ago a decisive contribution to the algebraic approach of the perturbation theory of periodic orbits of plane polynomial vector fields. This article presents first steps of a general framework in which a generalization of Bautin’s ideas to any dimension could be developed. The main result is the generalization of the algorithm of the successive derivatives of return mappings for 2-dimensional systems to any dimension in this framework