  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 if 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 dimension 5 that
is not definable in any o-minimal extension of the reals
