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Book Review
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Book Information
Author(s):
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:
- [1]
- R. Bamon, Quadratic vector fields in the plane have a finite number of limit cycles, Inst. Hautes \'Etudes Sci. Publ. Math. \textbf{64} (1986), 111--142.
- [2]
- C. Chicone and D. Shafer, Separatrix and limit cycles of quadratic systems and Dulac's theorem, Trans. Amer. Math. Soc. \textbf{278} (1983), 585--612.
- [3]
- A. Coppel, A survey of quadratic systems, J. Differential Equations \textbf{2} (1966), 293--304.
- [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.
- [5]
- H. Dulac, Sur les cycles limites, Bull. Soc. Math. France \textbf{51} (1923), 45--188.
- [6]
- F. Dumortier, Singularities of vector fields, Monografias de Math., vol. 32, IMPA, Rio de Janeiro, 1978.
- [7]
- F. Dumortier, R. Roussarie, and C. Rousseau, Hilbert's {\rm 16}th problem for quadratic vector fields, {\rm preprint 1992}.
- [8]
- J. \'Ecalle, J. Martinet, R. Moussu, and J-P. Ramis, Non-accumulation des cycles-limites {\rm I}, C. R. Acad. Sci. Paris S\'er. I Math. \textbf{304} (1987), 375--377.
- [9]
- J. \'Ecalle, Finitude des cycles limites it acc\'el\'ero-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 \textbf{8} (1902), 437--479.
- [11]
- Yu. S. Il\cprime yashenko, Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane, Funct. Anal. Appl. \textbf{18} (1984), 199--209.
- [12]
- Yu. S. Il\cprime yashenko, ``On limit cycles'' and related problems of the local theory of differential equations, Russian Math. Surveys \textbf{40} (1985), 1--49.
- [13]
- Yu. S. Il\cprime yashenko, The finiteness theorem for limit cycles, Uspekhi Mat. Nauk. \textbf{42} (1987), 223 \afterall (Russian).
- [14]
- Yu. S. Il\cprime yashenko, Finiteness theorems for limit cycles, Russian Math. Surveys \textbf{45} (1990), 129--203.
- [15]
- H. Poincar\'e, M\'emore sur les courbes d\'efinies par une equation diff\'erentielle, J. Math\'ematiques \textbf{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 \textbf{23} (1980), 153--158.
- [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.
Additional Information:
Reviewer(s):
Carmen
Chicone
Review Information:
Journal:
Bull. Amer. Math. Soc.
28
(1993),
123-130.
DOI:
10.1090/S0273-0979-1993-00329-X
PII:
S 0273-0979(1993)00329-X
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