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Book Review

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Book Information:

Author: Yu. S. Il\cprime yashenko
Title: Finiteness theorems for limit cycles
Additional book information: Translations of Math. Monographs, American Mathematical Society, Providence, RI, 1991, 288 + ix pp., US$196.00. ISBN 0-8218-4553-5.

References [Enhancements On Off] (What's this?)

  • [1] R. Bamon, Quadratic vector fields in the plane have a finite number of limit cycles, Inst. Hautes Études Sci. Publ. Math. 64 (1986), 111-142. MR 876161 (88d:58095)
  • [2] C. Chicone and D. Shafer, Separatrix and limit cycles of quadratic systems and Dulac's theorem, Trans. Amer. Math. Soc. 278 (1983), 585-612. MR 701513 (84m:58110)
  • [3] A. Coppel, A survey of quadratic systems, J. Differential Equations 2 (1966), 293-304. MR 0196182 (33:4374)
  • [4] S. Diliberto, On systems of ordinary differential equations, Contributions to the Theory of Nonlinear Oscillations I, Ann. of Math. Stud., vol. 20, Princeton Univ. Press, Princeton, NJ, 1950, pp. 1-38. MR 0034931 (11:665i)
  • [5] H. Dulac, Sur les cycles limites, Bull. Soc. Math. France 51 (1923), 45-188. MR 1504823
  • [6] F. Dumortier, Singularities of vector fields, Monografias de Math., vol. 32, IMPA, Rio de Janeiro, 1978. MR 526571 (81k:58050)
  • [7] F. Dumortier, R. Roussarie, and C. Rousseau, Hilbert's 16th problem for quadratic vector fields, preprint, 1992. MR 1275538 (95d:58095)
  • [8] J. Écalle, J. Martinet, R. Moussu, and J-P. Ramis, Non-accumulation des cycles-limites I, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), 375-377. MR 889742 (89i:58121a)
  • [9] J. Écalle, Finitude des cycles limites it accéléro-sommation de l'application de retour, Lecture Notes in Math., vol. 1455, Springer-Verlag, Berlin and New York, 1990, pp. 74-159.
  • [10] D. Hilbert, Mathematical problems, (M. Newton, transl.), Bull. Amer. Math. Soc 8 (1902), 437-479. MR 1557926
  • [11] Yu. S. Il'yashenko, Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane, Funct. Anal. Appl. 18 (1984), 199-209. MR 757247 (86a:34054)
  • [12] -, "On limit cycles" and related problems of the local theory of differential equations, Russian Math. Surveys 40 (1985), 1-49.
  • [13] -, The finiteness theorem for limit cycles, Uspekhi Mat. Nauk. 42 (1987), 223. (Russian)
  • [14] -, Finiteness theorems for limit cycles, Russian Math. Surveys 45 (1990), 129-203. MR 1069351 (92a:58110)
  • [15] H. Poincaré, Mémore sur les courbes définies par une equation différentielle, J. Mathématiques 7 (1881), 375-422.
  • [16] J. Reyn, A bibliography of the qualitative theory of quadratic systems of differential equations in the plane, TU Delft Technical Report 89-71, 1989.
  • [17] S-L. Shi, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica 23 (1980), 153-158. MR 574405 (81f:34037)
  • [18] J. Sotomayor and R. Paterlini, Quadratic vector fields with finitely many periodic orbits, Lecture Notes in Math., vol 1007, Springer-Verlag, Berlin and New York, 1983, pp. 753-766. MR 730297 (85b:58107)

Review Information:

Reviewer: Carmen Chicone
Journal: Bull. Amer. Math. Soc. 28 (1993), 123-130
DOI: https://doi.org/10.1090/S0273-0979-1993-00329-X
American Mathematical Society