Tame semiflows for piecewise linear vector fields
Annales de l'Institut Fourier, Volume 52 (2002) no. 6, pp. 1593-1628.

Let be a disjoint decomposition of n and let X be a vector field on n , defined to be linear on each cell of the decomposition . Under some natural assumptions, we show how to associate a semiflow to X and prove that such semiflow belongs to the o-minimal structure an ,exp . In particular, when X is a continuous vector field and Γ is an invariant subset of X, our result implies that if Γ is non-spiralling then the Poincaré first return map associated Γ is also in an ,exp .

Soit une décomposition disjointe de n et soit X un champ de vecteurs sur n , défini comme étant linéaire sur chaque cellule de la décomposition . Sous certaines hypothèses naturelles, nous montrons comment associer un semi-flot à X et nous montrons qu’un tel semi-flot appartient à la structure o-minimale an ,exp . En particulier, si X est un champ de vecteurs continu et Γ est un sous-ensemble invariant par X, notre résultat implique que l’application de premier retour de Poincaré associée à Γ est également dans an ,exp quand Γ est non-spiralante.

DOI: 10.5802/aif.1928
Classification: 03C64,  14P10,  34C25,  37G15
Keywords: piecewise linear vector field, o-minimal, semiflow
Panazzolo, Daniel 1

1 Universidade de São Paulo, Dep. Matemática Aplicada, Rua do Matao, 1010, São Paulo 05508-090 (Brésil)
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Panazzolo, Daniel. Tame semiflows for piecewise linear vector fields. Annales de l'Institut Fourier, Volume 52 (2002) no. 6, pp. 1593-1628. doi : 10.5802/aif.1928. https://aif.centre-mersenne.org/articles/10.5802/aif.1928/

[A] Arnold V.I. Ordinary Differential Equations, Moscow, 1984

[ACT] Arneodo A.; Coullet P.; Tresser C. Oscillators with chaotic behavior: an illustration of a theorem by Shilnikov, J. Statist. Phys., Volume 27 (1982) no. 1, pp. 171-182 | DOI | MR | Zbl

[BH] Bhatia N.P.; Hajek O. Local Semi-Dynamical Systems, Lecture Note in Math., Volume 90 (1969) | MR | Zbl

[BR] Benedetti R.; Risler J.-J. Real Algebraic and semi-algebraic sets, Hermann, Paris, 1990 | MR | Zbl

[D] van den Dries L. Tame Topology and O-minimal Structures, Cambridge University Press, 1998 | MR | Zbl

[DMM] van den Dries L.; Macintyre A.; Marker D. The elementary theory of restricted analytic fields with exponentiation, Ann. of Math., Volume 140 (1994) no. 2, pp. 183-205 | DOI | MR | Zbl

[F] Fulmer EP. Computation of the Matrix Exponential, Amer. Math. Monthly, Volume 82 (1975), pp. 156-159 | DOI | MR | Zbl

[Fi] Filippov A.F. Differential equations with discontinuous right-hand side, Mat. Sb. (N.S., Russian), Volume 51 (1960), pp. 99-128 | MR | Zbl

[Ha] Hájek O. Discontinuous Differential Equations I, J. Diff. Eq., Volume 32 (1979), pp. 149-170 | DOI | MR | Zbl

[Hi] Hirsch M. Differential Topology, Springer-Verlag, 1976 | MR | Zbl

[Ka] Kaloshin V. The Hilbert 16th Problem and an Estimate for Cyclicity of an Elementary Polycycle (Preprint)

[Kh] Khovanskii A.G. Fewnomials, AMS Translations of Mathematical Monographs, Volume 88 | MR | Zbl

[LR] Lion J.-M.; Rolin J.-P. Théorème de préparation pour les fonctions logarithmico-exponentielles, Ann. Inst. Fourier, Volume 47 (1997) no. 3, pp. 859-884 | DOI | Numdam | MR | Zbl

[M] Maxwell S. A general model completeness result for expansions of the real ordered field, Ann. Pure Appl. Logic, Volume 95 (1998) no. (1-3), pp. 185-227 | DOI | MR | Zbl

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

[S] Shilnikov L. Chua's circuit: rigorous results and future problems, Internat. J. Bifur. Chaos, Volume 4 (1994) no. 3, pp. 489-519 | DOI | MR

[W] Wilkie A.J. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc., Volume 9 (1996), pp. 1051-1094 | DOI | MR | Zbl

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