Poincare normal forms and Lie point symmetries

Abstract

We study Poincare normal forms of vector fields in the presence of symmetry under general-i.e. not necessarily linear-diffeomorphisms. We show that it is possible to reduce both the vector field and the symmetry diffeomorphism to normal form by means of an algorithmic procedure similar to the usual one for Poincare normal forms without symmetry; this 'joint' normal form can be given a simple geometric characterization.