Quasi‐analytic solutions of analytic ordinary differential equations and o‐minimal structures

It is well known that the non‐spiraling leaves of real analytic foliations of codimension 1 all belong to the same o‐minimal structure. Naturally, the question arises of whether the same statement is true for non‐oscillating trajectories of real analytic vector fields. We show, under certain assumptions, that such a trajectory generates an o‐minimal and model‐complete structure together with the analytic functions. The proof uses the asymptotic theory of irregular singular ordinary differential equations in order to establish a quasi‐analyticity result from which the main theorem follows. As applications, we present an infinite family of o‐minimal structures such that any two of them do not admit a common extension, and we construct a non‐oscillating trajectory of a real analytic vector field in ℝ 5 that is not definable in any o‐minimal extension of ℝ.