Let (S) : (y,z)' = F(x,y,z) be a system of two ordinary differential equations, with F definable in a polynomially bounded o-minimal structure. We introduce a notion of regular separation for the solutions of (S), inspired by a similar notion dur to Lojasiewicz. We then show a dichotomy for the behavior of couples of solutions of (S) which have flat contact, assuming one of them has regular separation: either they spiral infinitely many time the one around the other (they are interlaced), or else they both belong to a common Hardy field.

(This is a joint work with M. Matusinski and F. Sanz)