RT info:eu-repo/semantics/article
T1 Solutions of definable ODEs with regular separation and dichotomy interlacement versus Hardy
A1 Le Gal, Olivier
A1 Matusinski, Mickaël
A1 Sanz Sánchez, Fernando
K1 solutions of ODEs, non-oscillating trajectories of vector fields, o-minimality, Hardy field, transcendental formal solutions
AB We introduce a notion of regular separation for solutions of systems ofODEs y'=F(x; y), where F is definable in a polynomially bounded o-minimalstructure 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 interlacedor generates a Hardy field. We adapt this result to trajectories of three-dimensionalvector fields with definable coefficients. In the particular case of real analytic vectorfields, 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 thatthe set of trajectories with the regular separation property and asymptotic to a formalinvariant curve is never empty and it is represented by a subanalytic set of minimaldimension containing the curve. Finally, we show how to construct examplesof 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 thethird author.
PB European Mathematical Society Press
SN 0213-2230
YR 2021
FD 2021
LK https://uvadoc.uva.es/handle/10324/68129
UL https://uvadoc.uva.es/handle/10324/68129
LA eng
NO Rev. Mat. Iberoam. 38 (2022), no. 5, 1501–1527
NO Producción Científica
DS UVaDOC
RD 11-jul-2024