Book Review
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MathSciNet review:
1568001
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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.
Rodrigo Bamón, 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
Carmen Chicone and Douglas S. Shafer, Separatrix and limit cycles of quadratic systems and Dulac’s theorem, Trans. Amer. Math. Soc. 278 (1983), no. 2, 585–612. MR 701513, DOI 10.1090/S0002-9947-1983-0701513-X
W. A. Coppel, A survey of quadratic systems, J. Differential Equations 2 (1966), 293–304. MR 196182, DOI 10.1016/0022-0396(66)90070-2
Stephen P. Diliberto, On systems of ordinary differential equations, Contributions to the Theory of Nonlinear Oscillations, Annals of Mathematics Studies, no. 20, Princeton University Press, Princeton, N.J., 1950, pp. 1–38. MR 0034931
H. Dulac, Sur les cycles limites, Bull. Soc. Math. France 51 (1923), 45–188 (French). MR 1504823
Freddy Dumortier, Singularities of vector fields, Monografías de Matemática [Mathematical Monographs], vol. 32, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, 1978. MR 526571
F. Dumortier, R. Roussarie, and C. Rousseau, Elementary graphics of cyclicity $1$ and $2$, Nonlinearity 7 (1994), no. 3, 1001–1043. MR 1275538
Jean Écalle, Jean Martinet, Robert Moussu, and Jean-Pierre Ramis, Non-accumulation des cycles-limites. I, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 13, 375–377 (French, with English summary). MR 889742
[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.
David Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 8 (1902), no. 10, 437–479. MR 1557926, DOI 10.1090/S0002-9904-1902-00923-3
Yu. S. Il′yashenko, Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane, Funktsional. Anal. i Prilozhen. 18 (1984), no. 3, 32–42 (Russian). MR 757247
[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)
Yu. S. Il′yashenko, Finiteness theorems for limit cycles, Uspekhi Mat. Nauk 45 (1990), no. 2(272), 143–200, 240 (Russian); English transl., Russian Math. Surveys 45 (1990), no. 2, 129–203. MR 1069351, DOI 10.1070/RM1990v045n02ABEH002335
[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.
Song Ling Shi, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica 23 (1980), no. 2, 153–158. MR 574405
J. Sotomayor and R. Paterlini, Quadratic vector fields with finitely many periodic orbits, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 753–766. MR 730297, DOI 10.1007/BFb0061444
- [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
