In this talk we present the following result obtained jointly with O. Le Gal, that generalizes to any dimension a result by F. Dumortier and P. Bonckaert in 1986 concerning the realizability of formal invariant curves of three-dimensional vector fields:

Let $\xi$ be a $C^{\infty}$ vector field at $(\mathbb R^{n},0)$ and suppose that it has a formal invariant curve $\Gamma$. Then, as soon as the Taylor expansion of $\xi$ is not identically zero along $\Gamma$ (a necessary condition), there is a trajectory $\gamma\subset \mathbb R^{n}$ of $\xi$ which has $\Gamma$ as an asymptotic expansion at the origin. In fact, we realize the family of all trajectories which are asymptotic to $\Gamma$: we construct an invariant $C^0$ manifold in some open horn around $\Gamma$, entirely composed of asymptotic trajectories, and containing the germ of any such trajectory.

Furthermore, 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.