We introduce a notion of regular separation for solutions of systems of
ODEs y'=F(x; y), where F is definable in a polynomially bounded o-minimal
structure and y=(y1,y2). Given a pair of solutions with flat contact, we prove that,
if one of them has the property of regular separation, the pair is either interlaced
or generates a Hardy field. We adapt this result to trajectories of three-dimensional
vector fields with definable coefficients. In the particular case of real analytic vector
fields, it improves the dichotomy interlaced/separated of certain integral pencils,
obtained by F. Cano, R. Moussu and the third author. In this context, we show that
the set of trajectories with the regular separation property and asymptotic to a formal
invariant curve is never empty and it is represented by a subanalytic set of minimal
dimension containing the curve. Finally, we show how to construct examples
of formal invariant curves which are transcendental with respect to subanalytic sets,
using the so-called (SAT) property, introduced by J.-P. Rolin, R. Shaefke and the
third author.
