Quasi-linearization of parameter-depending germs of vector fields

摘要

In this paper, we study normalizationand quasi-linearization of a family of germs of hyperbolic vector fields at the origin. We show that, when the eigenvalues of these systems satisfy the so-called strongly 1-resonant condition, i.e. there exists a relation of the form (r, λ) = 0, then they can be simplified, in the context of orbital equivalence, to a normal form which can be integrated in an explicit and convenient way. More precisely, given a family of germs of sufficiently smooth vector fields X_ε having a generic nonlinear part, with X_0 strongly 1-resonant, then for any integer k, there is a neighbourhood V at the origin of the parameter space such that for any ε ∈ V, X_ε is C~k equivalent to a system having only one resonant term on each component.