Le problème de la finitude du nombre de cycles limites

Robert Moussu

Séminaire Bourbaki (1985-1986)

  • Volume: 28, page 89-101
  • ISSN: 0303-1179

How to cite

top

Moussu, Robert. "Le problème de la finitude du nombre de cycles limites." Séminaire Bourbaki 28 (1985-1986): 89-101. <http://eudml.org/doc/110073>.

@article{Moussu1985-1986,
author = {Moussu, Robert},
journal = {Séminaire Bourbaki},
keywords = {limit cycles; polycycles; vector field; Hilbert's sixteenth problem},
language = {fre},
pages = {89-101},
publisher = {Société Mathématique de France},
title = {Le problème de la finitude du nombre de cycles limites},
url = {http://eudml.org/doc/110073},
volume = {28},
year = {1985-1986},
}

TY - JOUR
AU - Moussu, Robert
TI - Le problème de la finitude du nombre de cycles limites
JO - Séminaire Bourbaki
PY - 1985-1986
PB - Société Mathématique de France
VL - 28
SP - 89
EP - 101
LA - fre
KW - limit cycles; polycycles; vector field; Hilbert's sixteenth problem
UR - http://eudml.org/doc/110073
ER -

References

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.