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

TL;DR

This paper introduces a new notion of regular separation for solutions of definable ODEs, proving a dichotomy between interlaced solutions or those generating Hardy fields, with applications to vector fields and invariant curves.

Contribution

It establishes a novel regular separation concept for definable ODE solutions and extends the interlaced/separated dichotomy to broader contexts, including Hardy fields and invariant curves.

Findings

Solutions with regular separation are either interlaced or generate Hardy fields.

The set of trajectories asymptotic to a formal invariant curve with regular separation is non-empty and subanalytic.

Constructs examples of transcendental invariant curves using the (SAT) property.

Abstract

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 $y = (y_1,y_2)$. 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.