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Quadratic systems with a weak focus

Published online by Cambridge University Press:  17 April 2009

Zhang Pingguang  and
Cai Suilin
Zhang Pingguang
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou Zhejiang, Peoples Republic of China
Cai Suilin
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou Zhejiang, Peoples Republic of China
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Abstract

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In this paper we study the number and the relative position of the limit cycles of a plane quadratic system with a weak focus. In particular, we prove the limit cycles of such a system can never have (2, 2)-distribution, and that there is at most one limit cycle not surrounding this weak focus under any one of the following conditions:

(i) the system has at least 2 saddles in the finite plane,

(ii) the system has more than 2 finite singular points and more than 1 singular point at infinity,

(iii) the system has exactly 2 finite singular points, more than 1 singular point at infinity, and the weak focus is itself surrounded by at least one limit cycle.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 44 , Issue 3 , December 1991 , pp. 511 - 526
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Andronov, A.A., Leontovich, E.A., Gordon, I.I. and Maier, A.G., Qualitative theory of second-order dynamic systems, English translation (Halsted Press, New York, 1973).Google Scholar
[2]Čerkas, L.A., ‘The absence of limit cycles for a certain differential equation that has a structurally unstable focus’, (Russian), Differentsial'nye Uravneniya 6 (1970), 779–783.Google Scholar
[3]Coppel, W.A., ‘Some quadratic systems with at most one limit cycle’, Dynamics Reported 2 (1989), 61–88.CrossRefGoogle Scholar
[4]Qin, Y.X., Shi, S.L. and Cai, S.L., ‘On limit cycles of planar quadratic system’, Sci. Sinica Ser. A 25 (1982), 41–50.Google Scholar
[5]Qin, Y.X., Suo, G.G. and Du, X.F., ‘On limit cycles of planar quadratic system’, Sci. Sinica Ser. A 26 (1983), 1025–1033.Google Scholar
[6]Shi, S.L., ‘On limit cycles of plane quadratic systems’, Sci. Sinico 24 (1981), 153–159.Google Scholar
[7]Ye, Y.Q., Cai, S.L., Chen, L.S., Huang, K.C., Luo, D.J., Ma, Z.E., Wang, E.N., Wang, M.S. and Yang, X.A., Theory of limit cycles, Translation of Mathematical Monographs 66 (American Mathematical Society, Providence, R.I., 1986).Google Scholar
[8]Zhang, P.G. and Li, W.H., ‘A quadratic system with weak focus and strong focus’, (preprint).Google Scholar