Bulletin of the Australian Mathematical Society

Table of Contents - Volume 44 - Issue03

Research Article

Quadratic systems with a weak focus

Zhang Pingguanga1 and Cai Suilina1

a1 Department of Mathematics, Zhejiang University, Hangzhou Zhejiang, Peoples Republic of China

Abstract

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.

(Received December 21 1990)

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. [OpenURL Query Data]  [Google Scholar]
  • [3] Coppel, W.A., ‘Some quadratic systems with at most one limit cycle’, Dynamics Reported 2 (1989), 61–88. [OpenURL Query Data]  [Google 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. [OpenURL Query Data]  [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. [OpenURL Query Data]  [Google Scholar]
  • [6] Shi, S.L., ‘On limit cycles of plane quadratic systems’, Sci. Sinico 24 (1981), 153–159. [OpenURL Query Data]  [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).