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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 structureand $y = (y_1,y_2)$. Given a pair of solutions with flat contact, we provethat, if one of them has the property of regular separation, the pair is eitherinterlaced or generates a Hardy field. We adapt this result to trajectories ofthree-dimensional vector fields with definable coefficients. In the particularcase of real analytic vector fields, it improves the dichotomyinterlaced/separated of certain integral pencils obtained by F. Cano, R. Moussuand the third author. In this context, we show that the set of trajectorieswith the regular separation property and asymptotic to a formal invariant curveis never empty and it is represented by a subanalytic set of minimal dimensioncontaining the curve. Finally, we show how to construct examples of formalinvariant curves which are transcendental with respect to subanalytic sets,using the so-called (SAT) property introduced by J.-P. Rolin, R. Shaefke andthe third author.

Solutions of definable ODEs with regular separation and dichotomy interlacement versus Hardy