Quadratic vector fields in the plane have a finite number of limit cycles

Rodrigo Bamon

Publications Mathématiques de l'IHÉS (1986)

  • Volume: 64, page 111-142
  • ISSN: 0073-8301

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Bamon, Rodrigo. "Quadratic vector fields in the plane have a finite number of limit cycles." Publications Mathématiques de l'IHÉS 64 (1986): 111-142. <http://eudml.org/doc/104014>.

@article{Bamon1986,
author = {Bamon, Rodrigo},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {Hilbert's 16-th problem; limit cycles; isolated periodic orbits; hyperbolic graphs; quadratic vector fields},
language = {eng},
pages = {111-142},
publisher = {Institut des Hautes Études Scientifiques},
title = {Quadratic vector fields in the plane have a finite number of limit cycles},
url = {http://eudml.org/doc/104014},
volume = {64},
year = {1986},
}

TY - JOUR
AU - Bamon, Rodrigo
TI - Quadratic vector fields in the plane have a finite number of limit cycles
JO - Publications Mathématiques de l'IHÉS
PY - 1986
PB - Institut des Hautes Études Scientifiques
VL - 64
SP - 111
EP - 142
LA - eng
KW - Hilbert's 16-th problem; limit cycles; isolated periodic orbits; hyperbolic graphs; quadratic vector fields
UR - http://eudml.org/doc/104014
ER -

References

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  1. [A] A. ANDRONOV et al., Qualitative Theory of Second Order Dynamical Systems, John Wiley & Sons, New York, 1973. Zbl0282.34022MR50 #2619
  2. [C] W. COPPEL, A survey of Quadratic Systems, Journal of Differential Equations, 2 (1966), 293-304. Zbl0143.11903MR33 #4374
  3. [Ca] J. CARR, Applications of Center Manifold Theory, Applied Math. Sciences, 35, Springer-Verlag, 1981. Zbl0464.58001MR83g:34039
  4. [Ch-S] C. CHICONE and S. SCHAFER, Separatrix and Limit Cycles of Quadratic Systems and Dulac's Theorem, Transactions Amer. Math. Soc., 278 (1983), 585-612. Zbl0522.58041MR84m:58110
  5. [D] M. H. DULAC, Sur les cycles limites, Bull. Soc. Math. France, 51 (1923), 45-188. JFM49.0304.01
  6. [Du] F. DUMORTIER, Singularities of Vector Fields, Journal of Differential Equations, 23, I (1977), 53-106. Zbl0346.58002MR58 #31276
  7. [H-P-S] M. HIRSCH, C. PUGH and M. SHUB, Invariant Manifolds, Springer Lecture Notes in Math., 583 (1977). Zbl0355.58009MR58 #18595
  8. [I] Yu. S. Il'yaŠenko, Limit cycles of polynomial vector fields with non degenerate singular points in the real plane (in Russian), Functional Analysis and its applications, 18 (3) (1984), 32-34 (in translation : 18 (3) (1985), 199-209). Zbl0549.34033
  9. [P-L1] I. G. PETROVSKII and E. U. LANDIS, On the number of limit cycles of the equation dy/dx = P(x,y)/Q(x,y) where P and Q are polynomials of the second degree, Amer. Math. Soc. Transl. (2), 16 (1958), 177-221. Zbl0080.07502MR20 #1036
  10. [P-L2] I. G. PETROVSKII and E. U. LANDIS, On the number of limit cycles of the equation dy/dx = P(x,y)/Q(x,y) where P and Q are polynomials, Amer. Math. Soc. Transl. (2), 14 (1960), 181-200. Zbl0094.06304MR22 #3854
  11. [P-L3] I. G. PETROVSKII and E. U. LANDIS, Corrections to the articles : "On the number of limit cycles of the equation dy/dx = P(x,y)/Q(x,y) where P and Q are polynomials", Math. Sb.N.S., 48 (90) (1959). 253-255. MR23 #A1099
  12. [P-M] J. PALIS and W. de MELO, Geometric Theory of Dynamical Systems ; An Introduction, New York, Springer-Verlag, 1982. Zbl0491.58001MR84a:58004
  13. [S] J. SOTOMAYOR, Curvas definidas por equaçoes diferenciais no plano, 13° Colóquio Bras. de Mat., IMPA, 1981. MR84m:34004
  14. [Sh1] SHI SONGLING, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci., Sinica, Ser. A, 23 (1980), 153-158. Zbl0431.34024MR81f:34037
  15. [Sh2] SHI SONGLING, A method for constructing cycles without contact around a weak focus, Journal of Differential Equations, 41 (1981), 301-312. Zbl0442.34029MR83e:58074

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