Parametric normal forms of vector fields and their further simplification

Abstract

Given a family of vector fields parametrized by ξ in its linear part, one usually obtains its versal unfolding by the so-called two-step approach, i.e. find a Poincaré normal form for the system with fixed ξ0 and then apply the obtained near-identity transformation to the system with general ξ. It is also a common practice to treat ξ as a component together with those state components and calculate normal forms of the extended system. In this paper we reformulate normal forms on modules of homogeneous polynomials over the ring of all continuous functions of ξ and give a direct computation of versal unfolding. Our procedure enables us to determine coefficients of all terms of a certain degree in the normal form before we give a near-identity transformation of this degree. We can give all available near-identity transformations and choose an appropriate one to eliminate more terms of higher degree for a simpler normal form. We prove that the normal form reduced in our procedure is the simplest and unique. We illustrate our method with systems of linear centre and nilpotent linear parts separately.