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Título
Solutions of definable ODEs with regular separation and dichotomy interlacement versus Hardy
Año del Documento
2021
Editorial
European Mathematical Society Press
Descripción
Producción Científica
Documento Fuente
Rev. Mat. Iberoam. 38 (2022), no. 5, 1501–1527
Resumen
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=(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 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.
Palabras Clave
solutions of ODEs, non-oscillating trajectories of vector fields, o-minimality, Hardy field, transcendental formal solutions
ISSN
0213-2230
Revisión por pares
SI
DOI
Patrocinador
F. Sanz Sánchez was partially supported by Ministerio de Ciencia, Spain, process MTM2016-77642-C2-1-P and PID2019-105621GB-I00
Propietario de los Derechos
Real Sociedad Matemática Española
Idioma
eng
Tipo de versión
info:eu-repo/semantics/submittedVersion
Derechos
restrictedAccess
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