Solutions non oscillantes d’une équation différentielle et corps de Hardy

Blais, François; Moussu, Robert; Sanz, Fernando

Annales de l'Institut Fourier, Tome 57 (2007) no. 6, pp. 1825-1838.

Soit $\varphi :x\mapsto \varphi \left(x\right),x\gg 0$ une solution à l’infini d’une équation différentielle algébrique d’ordre $n$, $P(x,y,y^{\prime },...,y^{\left(n\right)})=0$. Nous donnons un critère géométrique pour que les germes à l’infini de $\varphi$ et de la fonction identité sur $ℝ$ appartiennent à un même corps de Hardy. Ce critère repose sur le concept de non oscillation.

Let $\varphi :x\mapsto \varphi \left(x\right),x\gg 0$ be a solution of an algebraic differential equation of order $n$, $P(x,y,y^{\prime },...,y^{\left(n\right)})=0$. We establish a geometric criterion so that the germs at infinity of $\varphi$ and the identity function on $ℝ$ belong to a common Hardy field. This criterion is based on the concept of non oscillation.

[1] Arnolʼd, V. Chapitres supplémentaires de la théorie des équations différentielles ordinaires, “Mir”, Moscow, 1984 (Translated from the Russian by Djilali Embarek, Reprint of the 1980 edition) | Zbl

[2] Benedetti, Riccardo; Risler, Jean-Jacques Real algebraic and semi-algebraic sets, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990 | MR | Zbl

[3] Bochnak, J.; Coste, M.; Roy, M.-F. Géométrie algébrique réelle, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 12, Springer-Verlag, Berlin, 1987 | MR | Zbl

[4] Boshernitzan, Michael An extension of Hardy’s class $L$ of “orders of infinity”, J. Analyse Math., Volume 39 (1981), pp. 235-255 | DOI | Zbl

[5] Boshernitzan, Michael Second order differential equations over Hardy fields, J. London Math. Soc. (2), Volume 35 (1987) no. 1, pp. 109-120 | DOI | MR | Zbl

[6] Cano, F.; Moussu, R.; Sanz, F. Oscillation, spiralement, tourbillonnement, Comment. Math. Helv., Volume 75 (2000) no. 2, pp. 284-318 | DOI | MR | Zbl

[7] Cano, F.; Moussu, R.; Sanz, F. Nonoscillating projections for trajectories of vector fields, Journal of Dynamical and Control Systems, Volume 13 (2007) no. 2, pp. 173-176 | DOI | MR | Zbl

[8] van den Dries, Lou Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, 248, Cambridge University Press, Cambridge, 1998 | MR | Zbl

[9] Grigorʼev, D. Yu.; Singer, M. F. Solving ordinary differential equations in terms of series with real exponents, Trans. Amer. Math. Soc., Volume 327 (1991) no. 1, pp. 329-351 | DOI | MR | Zbl

[10] Hardy, G. H. Properties of logarithmico-exponential functions, Proc. London Math. Soc.,, Volume 10 (1912) no. 2, pp. 54-90 | DOI

[11] Hardy, G. H. Some results concerning the behaviour at infinity of a real and continuous solution of an algebraic differential equation of the first order, Proc. London Math. Soc.,, Volume 10 (1912), pp. 451-469 | DOI

[12] Khovanskiĭ, A. G. Fewnomials, Translations of Mathematical Monographs, 88, American Mathematical Society, Providence, RI, 1991 (Translated from the Russian by Smilka Zdravkovska) | MR | Zbl

[13] Lion, Jean-Marie; Miller, Chris; Speissegger, Patrick Differential equations over polynomially bounded o-minimal structures, Proc. Amer. Math. Soc., Volume 131 (2003) no. 1, p. 175-183 (electronic) | DOI | MR | Zbl

[14] Moussu, R.; Roche, C. Théorie de Hovanskiĭ et problème de Dulac, Invent. Math., Volume 105 (1991) no. 2, pp. 431-441 | DOI | MR | Zbl

[15] Novikov, D.; Yakovenko, S. Trajectories of polynomial vector fields and ascending chains of polynomial ideals, Ann. Inst. Fourier (Grenoble), Volume 49 (1999) no. 2, pp. 563-609 | DOI | Numdam | MR | Zbl

[16] Perron, O. Über Differentialgliechungen erster Ordnung, die nicht nach der Ableitung aufgelöst sind, Jahresbericht der Deutschen Mathematiker-Vereinigung, Volume 22 (1912), pp. 356-368

[17] Rosenlicht, Maxwell Hardy fields, J. Math. Anal. Appl., Volume 93 (1983) no. 2, pp. 297-311 | DOI | MR | Zbl

[18] Rosenlicht, Maxwell Growth properties of functions in Hardy fields, Trans. Amer. Math. Soc., Volume 299 (1987) no. 1, pp. 261-272 | DOI | MR | Zbl

[19] Rosenlicht, Maxwell Asymptotic solutions of $Y^{\prime \prime }=F\left(x\right)Y$, J. Math. Anal. Appl., Volume 189 (1995) no. 3, pp. 640-650 | DOI | MR | Zbl

[20] Shackell, John Erratum : “Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations” [Ann. Inst. Fourier (Grenoble) 45 (1995), no. 1, 183–221 ; MR1324130 (96f :34073)], Ann. Inst. Fourier (Grenoble), Volume 45 (1995) no. 5, pp. 1471

[21] Valiron, Georges Équations Fonctionnelles. Applications, Masson et Cie, Paris, 1945 | MR | Zbl