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On the accumulation of limit cycles. (English) Zbl 0626.34022

Author’s abstract: Limit cycles of a planar analytic system can only accumulate at a critical point or on a graphic; the critical points are isolated; and the inner and outer boundaries of any continuous band of cycles consist of either a critical point or graphic.
Reviewer: V.Sree Hari Rao

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
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References:

[1] H. Dulac, Sur les cycles limites, Bull. Soc. Math. France 51 (1923), 45 – 188 (French). · JFM 49.0304.01
[2] 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. · Zbl 0522.58041
[3] H. Poincaré, Mémoire sur les courbes définies par une equation différentielle, J. Mathématiques 7 (1881), 375-422; Oeuvre (1880-1890), Gauthier-Villar, Paris, pp. 1-221.
[4] J. P. Françoise and C. C. Pugh, Keeping track of limit cycles, Berkeley Report, PAM-239, 1984. · Zbl 0602.34019
[5] Ivar Bendixson, Sur les courbes définies par des équations différentielles, Acta Math. 24 (1901), no. 1, 1 – 88 (French). · JFM 31.0328.03 · doi:10.1007/BF02403068
[6] G. F. D. Duff, Limit-cycles and rotated vector fields, Ann. of Math. (2) 57 (1953), 15 – 31. · Zbl 0050.09103 · doi:10.2307/1969724
[7] Salomon Bochner and William Ted Martin, Several Complex Variables, Princeton Mathematical Series, vol. 10, Princeton University Press, Princeton, N. J., 1948. · Zbl 0041.05205
[8] A. A. Andronov et al., Theory of bifurcations of dynamical systems on a plane, Kefer Press, Jerusalem, 1971.
[9] Lawrence M. Perko and Shih Lung Shü, Existence, uniqueness, and nonexistence of limit cycles for a class of quadratic systems in the plane, J. Differential Equations 53 (1984), no. 2, 146 – 171. · Zbl 0493.34025 · doi:10.1016/0022-0396(84)90037-8
[10] Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. · Zbl 0064.33002
[11] Morris W. Hirsch and Stephen Smale, Differential equations, dynamical systems, and linear algebra, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Pure and Applied Mathematics, Vol. 60. · Zbl 0309.34001
[12] A. A. Andronov et al., Qualitative theory of second order dynamical systems, Wiley, New York, 1973.
[13] L. M. Perko, Center-foci for analytic systems (in preparation).
[14] Yu. S. Il\(^{\prime}\)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). · Zbl 0549.34033
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