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Asymptotic properties of the Dulac map near a hyperbolic saddle in dimension three. (English. French summary) Zbl 1017.37009
Summary: We study the Dulac map associated to a vector field defined in a neighborhood of a hyperbolic equilibrium point of saddle type i.e. the map from a section transverse to the 2-dimensional stable manifold to another section transverse to the unstable one. We give an asymptotic expression of this Dulac map. We show that, this asymptotic expression has the same expression, up to some multiplication by a function of Mourtada type, as in the case where the vector field is linear.

MSC:
37C10Vector fields, flows, ordinary differential equations
37C75Stability theory
37D05Hyperbolic orbits and sets
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[1] Bonckaert ( P. ). - On the continuous dependence of the smooth change of coordinates in parametrized normal forms theorems , Journ. Diff. Eq. 106 ( 1993 ), 107 - 120 . MR 1249179 | Zbl 0788.58048 · Zbl 0788.58048 · doi:10.1006/jdeq.1993.1101
[2] Bonckaert ( P. ). - Conjugacy of vector fields respecting additional properties , Journal of Dynamical and Control Systems 3 ( 1997 ),pp. 419 - 432 . MR 1472359 | Zbl 01431839 · Zbl 1157.37322 · doi:10.1007/BF02463258
[3] Dumortier ( F. ) and Roussarie ( R. ). - On the saddle loop bifurcation . In Bifurcations of planar vector fields, proceedings Luminy 1989 , volume 1455 of Lecture Notes in Mathematics , pages 44 - 73 . Springer-Verlag , 1990 . MR 1094377 | Zbl 0731.34010 · Zbl 0731.34010
[4] Ilyashenko ( Y. ) and Yakovenko ( S. ). - Finite cyclicity of elementary polycycles . In Y. Ilyashenko and S. Yakovenko, editors, Concerning the Hilbert 16th Problem volume 165 of American Mathematical Society Translations Series 2 , pages 21 - 95 . American Mathematical Society , 1995 . MR 1334340 | Zbl 0828.34021 · Zbl 0828.34021
[5] IL’YASHENKO ( Yu.S. ) and Yakovenko ( S.Yu. ). - Finitely-smooth normal forms of local families of diffeomorphisms and vector fields , Russian Math. Surveys , 46 : 1 ( 1991 ), 1 - 43 . MR 1109035 | Zbl 0744.58006 · Zbl 0744.58006 · doi:10.1070/RM1991v046n01ABEH002733
[6] Homburg ( A.J. ), Kokubu ( H. ), Naudot ( V. ). - ” Homoclinic doubling cascades ”, ” Archive for Rational Mechanics and Analysis ”, 160 : 3 ( 2001 ), 195 - 244 . MR 1869442 | Zbl 1073.37059 · Zbl 1073.37059 · doi:10.1007/s002050100159
[7] Mourtada ( A. ). - Cyclicité finie des polycycles hyperboliques de champs de vecteurs du plan mise sous forme normale . In Bifurcations of planar vector fields, proceedings Luminy 1989 , volume 1455 of Lecture Notes in Mathematics , pages 272 - 314 . Springer-Verlag , 1990 . MR 1094384 | Zbl 0719.58031 · Zbl 0719.58031
[8] Moussu ( R. ). - Développement asymptotique de l’application retour d’un polycycle . In Dynamical Systems, Valparaiso 1986 , volume 1331 of Lecture Notes in Mathematics , pages 140 - 149 . Springer-Verlag , 1988 . MR 961097 | Zbl 0659.58009 · Zbl 0659.58009
[9] Roussarie ( R. ) On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields . Bol. Soc. Bras. Mat. , 17 : 67 - 101 , 1986 . MR 901596 | Zbl 0628.34032 · Zbl 0628.34032 · doi:10.1007/BF02584827
[10] Roussarie ( R. ) and Rousseau ( C. ). - Almost planar homoclinic loops in R3 . Journal of Differential Equations , 126 : 1 - 47 , 1996 . MR 1382055 | Zbl 0849.34036 · Zbl 0849.34036 · doi:10.1006/jdeq.1996.0042
[11] Rychlik ( M.R. ). - Lorenz attractors through Shil’nikov-type bifurcation . Part I, Ergod. Th. & Dynam. Syst. 10 ( 1990 ), 793 - 821 . Zbl 0715.58027 · Zbl 0715.58027 · doi:10.1017/S0143385700005915
[12] Sternberg ( S. ). - On the structure of local homomorphisms of euclidean n-space, I , Am. J. Math. 80 ( 1958 ), 623 - 31 . MR 96854 | Zbl 0083.31406 · Zbl 0083.31406 · doi:10.2307/2372774
[13] Sternberg ( S. ). - On the structure of local homomorphisms of euclidean n-space, II , Am. J. Math. 81 ( 1959 ), 578 - 605 . MR 109853 | Zbl 0211.56304 · Zbl 0211.56304 · doi:10.2307/2372915