Mickaël Matusinski (Institut de Mathématiques de Bordeaux) – Transserial trajectories of planar vector fields

In the context of a 3-dimensional real analytic vector field at a singular point, Cano, Moussu and Sanz introduced and studied the notion of integral pencils of trajectories at that point in order to obtain informations on the possible dynamical behaviours. We extend this approach on the formal side, taking advantage of the computability of transseries (in particular they are grid-based in the sense of Ecalle - van der Hoeven) when solving differential equations.

More precisely, for a real formal planar vector field at 0, we introduce a notion of transserial trajectories and provide an explicit description of all the possible transserial pencils. This is meant to be a first step toward the same sort of description in dimension 3.

As a motivation, we expect these transserial trajectories to reflect tameness properties of actual solutions, in a way similar to that of differentially algebraic transseries for germs in some Hardy fields: cf the recent results of Aschenbrenner-van den Dries-van der Hoeven.

Joint work in progress with Daniel Panazzolo and Fernando Sanz.