Le problème de la finitude du nombre de cycles limites

 Access to full text

 Full (PDF)

1. [1 ] R. Bamon - Solution of Dulac's problem for quadratic Vector Fields, Preprint IMPA, (1985). MR832735
2. [2 ] N.N. Bautin - On the number of limite cycles wich appear with the variation of coefficients from on equilibrium position of focus on centre type, Amer.Math. Soc. Transi., 100 , (1954). Zbl0059.08201MR59426
3. [3 ] N. Bourbaki - Les structures fondamentales de l'analyse, Fonction d'une variable réelle XII, livre IV, chapitre V et VI, Act. Scien. Indus., Hermann (1951), 55-159. 
4. [4 ] A.D. Bryuno - Analytical form of differential equations, Trans. Moscow. Math. Soc., 25, (1971), 131-288. Zbl0272.34018
5. [5 ] A.D. Bryuno - The normal form of real differential equations, Math. Notes, (1975), 722-731. Zbl0319.34010
6. [6 ] D. Cerveau & J.F. Mattei - Formes intégrables holomorphes singulières, Astérisque97 (1982). Zbl0545.32006MR704017
7. [7 ] C. Chicone & D.S. Shafer - Separatrix and limit cycles of quadratic systems and Dulac's theorem, Trans. Amer. Math. Soc. , Vol. 278, 2, (1983), 585-612. Zbl0522.58041MR701513
8. [8 ] W. Coppel - A survey of quadratic systems, J. of Diff. Eq., 2, (1966),293-304. Zbl0143.11903MR196182
9. [9 ] H. Dulac - Recherches sur les points singuliers des équations différentielles, J. Ecole Polytechnique, 2, 9, (1904), 1-25. JFM35.0331.02
10. [10 ] H. Dulac - Détermination et intégration d'une certaine classe d'équations différentielles ayant pour point singulier un centre, Bull. Sci. Math., 2, 32, (1908), 230-252. Zbl39.0374.01JFM39.0374.01
11. [11 ] H. Dulac - Sur les cycles limites, Bull. Soc. Math. France, 51, (1923), 45-188. Zbl49.0304.01MR1504823JFM49.0304.01
12. [12 ] F. Dumortier - Singularities of vector fields, J. of Diff. Eq., 23, 1, (1977). Zbl0346.58002MR650816
13. [13 ] F. Dumortier - Conférence à l'IMPA - RIO , (1976). 
14. [14 ] M. Frommer - Über das Auftreten von Wirbeln und Strudeln..., Math. Ann., 109, (1934), 395-424. Zbl0008.20703MR1512902JFM60.1094.01
15. [15 ] P. Hartman - Ordinary differential equations, Wiley, New-York, (1964). Zbl0125.32102MR171038
16. [16 ] D. Hilbert - Mathematical problems, Bull. Amer. Math. Soc., 8, (1902), 437-479. JFM33.0976.07
17. [17 ] A. Hovansky - Théorème de Bezout pour les fonctions de Liouville, Preprint M/81/45 - IHES, (1981). 
18. [18 ] Yu S. Il'yasenko - An example of equations dw/dz = Pn(z,w) /Qn(z,w) having..., Math. Sbor., 80, 3, (1969), 365-378. Zbl0214.09404
19. [19] Yu S. Il'yasenko - The finitness problem for limit cycles of polynomial vector fields on the plane..., Leningrad Int. Topo. Conf., abstracts, Nauka, Leningrad, (1982). 
20. [20] Yu S. Il'yasenko - Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane , Funk. Anal. Ego. Pri., 18, 3, (1984), 32-34.(Func. Ana. and Appl., 18,3, (1985), 199-209). Zbl0549.34033MR757247
21. [21] A.M. Lyapunov - Stability of motion, Academic Press , New-York, (1966). Zbl0161.06303MR208093
22. [22] J. Martinet - Lettre à Cerveau-Moussu , (1985) . 
23. [23] J. Martinet - Normalisation des champs de vecteurs holomorphes (d'après Bryuno), Séminaire Bourbaki, Exposé 564, Nov. 1980, L.N.901, Springer (1981), 55- 70. Zbl0481.34013MR647488
24. [24] J. Martinet & J.P. Ramis - Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Pub. Math. IHES, 55, (1982), 63- 164. Zbl0546.58038MR672182
25. [25] J. Martinet & J.P. Ramis - Classification analytique des équations différentielles non linéaires résonnantes du premier ordre, Ann. Sc. Ec. Norm.Sup., 4ème série, t.16 ,(1983), 571-625. Zbl0534.34011MR740592
26. [26] J. Martinet & J.P. Ramis - Analytic classification of resonnant saddles and foci, in singularities and dynamical systems, S.N.Pnevmatikos - Editor - North-Holland, (1985), 109-136. Zbl0596.34021MR806185
27. [27] J.F. Mattei & R. Moussu - Holonomie et integrales premières , Ann. Sc. Ec. Norm. Sup. , 13, (1980), 469-523. Zbl0458.32005MR608290
28. [28] R. Paterlini & J. Sotomayor - Quadratic vector fields with many periodic orbits, L.N., 1007, Springer, (1983), 753-766. Zbl0557.34038MR730297
29. [29] I.G. Petrovski & E.M. Landis - On the number of limit cycles of the equation dy/dx = P(x,y)/Q(x,y) where P and Q are polynomials, Am. Math. Soc. Transl., 2, 14, (1960), 181-200. Zbl0094.06304MR113013
30. [30] I.G. Petrovski & E.M. Landis - On the number of limit cycles of the equation dy/dx = P(x,y)/Q(x,y) where P are polynomials of the second degree, Am. Math. Soc. Transl.,2, 16, (1958), 177-221. Zbl0080.07502MR94521
31. [31] I.G. Petrovski & E.M. Landis - Correction to the articles "On the number...", Math. Sb. NS, 48, 90, (1965), 253-255. MR123777
32. [32] H. Poincare - Mémoire sur les courbes définies par une équation différentielle, J. Math., 7, (1881), 375-422. Zbl13.0591.01JFM13.0591.01
33. [33] J.P. Ramis - Les séries k-sommables et leurs applications, L.N. in Physics, 126, Springer, (1980). MR579749
34. [34] J.J. Risler - Complexité en géométrie réelle (d'après Khovansky), Séminaire Bourbaki - Exp. 637, (1984). Zbl0589.32020
35. [35] R. Roussarie - Modèles locaux de champs et de formes, Astérisque30, (1975). Zbl0327.57017MR440570
36. [36 ] A. Seidenberg - Reduction of singularities of the differentiable equation Ady = Bdx , Amer. J. Math. , (1968), 248-269. Zbl0159.33303MR220710
37. [37 ] Shi-Songling - A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica. Ser. A., 23, (1980), 153-158. Zbl0431.34024MR574405
38. [38 ] S. Sternberg - On the structure of local homeomorphisms of euclidean n-space II, Amer. J. Math, 80, (1958), 623-631. Zbl0083.31406MR96854
39. [39 ] J. Sotomayor - Curvas definidas por equacoes diferenciais no plano, 13ème col. Bras. Math., publi. IMPA, (1981). 
40. [40 ] A.N. Varchenko - Estimate of the number of zeros of an abelian integral depending on a parameter and limite cycles, Funct. Ana. and Appl.,18,3(1985), 98-108. Zbl0578.58035
41. [41] Ye Yan Quian - Some problems in quadratic theory of ordinary differential equations, J. of Diff. Eq., 46, 2, (1982). MR675905

1. R. Moussu, Problème de Dulac et variétés pfaffiennes
2. Jean-Christophe Yoccoz, Non-accumulation de cycles limites
3. Rémi Soufflet, Finiteness property for generalized abelian integrals