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Preprint Number 1894
1894. Olivier Le Gal, Mickaël Matusinski, Fernando Sanz Sánchez Solutions of definable ODEs with regular separation and dichotomy interlacement versus Hardy E-mail: Submission date: 13 December 2020 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 vs separated of certain integral pencils, as studied in [CMS04]. 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 from [RSS07]. Mathematics Subject Classification: 34C08, 34D05, 03C64 (Primary), 14P15, 32B20 (Secondary) Keywords and phrases: |
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