Abstract
Solutions of two-dimensional linear systems of ODEs with real meromorphic coefficients may have two very distinct kinds of relative behaviour when they approach to a singular point: either any two of them are linked or either any two of them can be separated by a linear projection. In this paper, we are interesting in the question of the decidability of the dichotomy linked/separated for the whole family of systems. First, we rewrite the known result which asserts that the dichotomy is determined in terms of a semialgebraic set (is decidable) on a truncation of the Taylor expansion of the coefficients of the system. After that, we study the question of the decidability of that dichotomy in terms of the coefficients of the system themselves as elements of the ordered Hardy field of real meromorphic functions.
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References
Balser, W.: Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations. Universitext. Springer, New York (2000)
Cano, F., Moussu, R., Sanz, F.: Oscillation, spiralement, tourbillonnement. Comment. Math. Helv. 75, 284–318 (2000)
Cano, F., Moussu, R., Sanz, F.: Pinceaux de courbes intégrales d’un champ de vecteurs analytique. Astérisque 297, 1–34 (2004)
Dieudonné, J.: Éléments d’analyse. Tome I: Fondements de l’analyse moderne. Gauthier-Villars, Éditeur, Paris (1968)
LeGal, O., Sanz, F., Speissegger, P.: Non-interlaced solutions of 2-dimensional systems of linear ordinary differential equations. Proc. Am. Math. Soc. 141, 2429–2438 (2013)
Miller, C.: Basics of O-minimality and Hardy Fields. In: Lecture Notes on O-minimal Structures and Real Analytic Geometry. Fields Institute Communications, vol. 62. Springer, New York (2012)
Wasow, W.: Asymptotic Expansions for Ordinary Differential Equations. Intersciencie, New York (1965) (Re-edited Dover, 1987)
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Carnicero, F.Á., Sanz, F. (2013). Interlacing and Separation of Solutions of Linear Meromorphic ODEs. In: Ibáñez, S., Pérez del Río, J., Pumariño, A., Rodríguez, J. (eds) Progress and Challenges in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 54. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38830-9_10
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DOI: https://doi.org/10.1007/978-3-642-38830-9_10
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