Abstract
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 having a generic nonlinear part, with strongly 1-resonant, then for any integer k, there is a neighbourhood at the origin of the parameter space such that for any , is C k equivalent to a system having only one resonant term on each component.
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