Trajectories of vector fields asymptotic to formal invariant curves

Abstract

We prove that a formal curve that is invariant by a C ∞ vector field ξ of Rm has a geometrical realization, as soon as the Taylor expansion of ξ is not identically zero along . This means that there is a trajectory γ ⊂ Rm of ξ which is asymptotic to . This result solves a natural question proposed by Bonckaert [Smooth invariant curves of singularities of vector fields in R3. Ann. Inst. Henri Poincaré 3(2) (1986), 111–183] nearly forty years ago. We also construct an invariant C0 manifold S in some open horn around which is composed entirely of trajectories asymptotic to and contains the germ of any such trajectory. If ξ is analytic, we prove that there exists a trajectory γ asymptotic to which is, moreover, non-oscillating with respect to subanalytic sets.