Quadratic systems having a parabola as an integral curve. (English) Zbl 0677.34034

Summary: The class of quadratic systems having a parabola composed of integral curves is examined. Canonical forms are found for the members of this class, and conditions are obtained, using the Bendixson’s criterion and the Poincaré-Bendixson theorem for the existence or non-existence of limit cycles, in the case where there is a limit cycle “inside” the parabola (that is, in the convex component of its complement).


34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
Full Text: DOI


[1] Chin, Sci Sinica Ser. A 7 pp 934– (1958)
[2] Chen, Kexue Tongbao 30 pp 401– (1985)
[3] Bautin, Mat. Sb. 30 pp 181– (1952)
[4] Yanqian, Theory of Limit Cycles 66 (1986)
[5] Čerkas, Differentsial’nye Uravneniya 13 pp 779– (1977)
[6] DOI: 10.1017/CBO9780511600777.007
[7] Guckenheimer, Degenerate homoclinic cycles in perturbations of quadratic Hamiltonian systems (1988)
[8] Coppel, Proceedings of the Ninth Conference on Ordinary and Partial Differential Equations, University of Dundee 157 pp 52– (1986)
[9] DOI: 10.1016/0022-0396(66)90070-2 · Zbl 0143.11903
[10] Yanqian, Qualitive theory of quadratic differential systems (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.