Trajectories of vector fields asymptotic to formal invariant curves

Abstract We prove that a formal curve $\Gamma $ that is invariant by a $C^{\infty }$ vector field $\xi $ of $\mathbb R^{m}$ has a geometrical realization, as soon as the Taylor expansion of $\xi $ is not identically zero along $\Gamma $ . This means that there is a trajectory $\gamma \subset \mathbb R^{m}$ of $\xi $ which is asymptotic to $\Gamma $ . This result solves a natural question proposed by Bonckaert [Smooth invariant curves of singularities of vector fields in R 3 . Ann. Inst. Henri Poincaré 3 (2) (1986), 111–183] nearly forty years ago. We also construct an invariant $C^0$ manifold S in some open horn around $\Gamma $ which is composed entirely of trajectories asymptotic to $\Gamma $ and contains the germ of any such trajectory. If $\xi $ is analytic, we prove that there exists a trajectory $\gamma $ asymptotic to $\Gamma $ which is, moreover, non-oscillating with respect to subanalytic sets.