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Quadratic systems having a parabola as an integral curve

Published online by Cambridge University Press:  14 November 2011

Colin Christopher
Colin Christopher
Affiliation:
Department of Mathematics, The University College of Wales, Aberystwyth SY23 3BZ, U.K.
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Synopsis

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 compliment).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 112 , Issue 1-2 , 1989 , pp. 113 - 134
Copyright
Copyright © Royal Society of Edinburgh 1989

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References

1.Bautin, N. N.. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus of centre type. Mat. Sb. 30 (1952), 181–196. Amer. Math. Soc. Transl. 100 (1954).Google Scholar
2.Chen, G. Q.. Real quadratic differential systems having a parabola as an integral curve. Proceedings of the 1983 Beijing Symposium on Differential Geometry and Differential Equations (1986).Google Scholar
3.Chen, S. P.. Limit cycles of a real quadratic differential system having a parabola as an integral curve. Kexue Tongbao 30 (1985) 401–405.Google Scholar
4.Chin, Yuan-Shun. On algebraic limit cycles of degree 2 of the differential equation dy/dx = . Sci Sinica Ser. A 7 (1958) 934–945.Google Scholar
5.Čerkas, L. A.. Methods for estimating the number of limit cycles for autonomous systems. Differentsial'nye Uravneniya 13 (1977) 779–802, 963. Differential Equations 13 (1978), 529–547.Google Scholar
6.Coppel, W. A.. A survey of quadratic systems. J. Differential Equations 2 (1966), 293–304.CrossRefGoogle Scholar
7.Coppel, W. A.. The limit cycle configurations of quadratic system. Proceedings of the Ninth Conference on Ordinary and Partial Differential Equations, University of Dundee, 1986, 52–65. Pitman Research Notes in Mathematics 157 (Harlow: Longman 1987).Google Scholar
8.Guckenheimer, J., Rand, R. and Schlomiuk, D.. Degenerate homoclinic cycles in perturbations of quadratic Hamiltonian systems (Preprint, Cornell University, 1988).Google Scholar
9.Lloyd, N. G.. Limit cycles of polynomial systems—some recent developments. In New Directions in Dynamical Systems, eds. Bedford, T. and Swift, J., pp. 192–234. L. M. S. Lecture Note Series 127 (Cambridge: Cambridge University Press, 1988).CrossRefGoogle Scholar
10.Suo, G. J.. Real quadratic differential systems having a parabola as an integral curve may have a limit cycle. Proceedings of the 1983 Beijing Symposium on Differential Geometry and Differential Equations (1986).Google Scholar
11.Yanqian, Ye. Qualitive theory of quadratic differential systems (Lectures given at the Institute de Recherche Mathèmatique Advancèe, University of Strasbourg, 1983–1984).Google Scholar
12.Yanqian, Yeet al. Theory of Limit Cycles. Trans. Math. Monographs 66 (Providence, R.I.: American, Mathematical Society, 1986).Google Scholar