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| Saddle Values and Integrability Conditions of Quadratic Differential Systems |
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Citation： |
Zhu Deming.Saddle Values and Integrability Conditions of Quadratic Differential Systems[J].Chinese Annals of Mathematics B,1987,8(4):466~478 |
| Page view： 787
Net amount： 962 |
Authors： |
Zhu Deming; |
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| Abstract： |
In the first section of the paper, the first three saddle values R_1,R_2,R_3 of the real quadratic differential system (QDE) are computed by use of the method with which Poincare researchs on Hopf bifurcation. In the second section, by applying the method and results of Dulac seeking integrability conditions of QDE it is proved that the system is integrable if and only if R_1=B_2=B_3=0, and it is also true when the system is complex. The integrability conditions given here can be applied much more easily then Dulac's. In the last part of the paper, it is pointed out that iR_1, iR_2, iR_3 are the first three focal values of the weak focus of the complex system. By the formulae of R_1, R_2,R_3 and the result in section 2, one can easily give the formulae of the focal values for the real QDE with Bautin form and give a new proof of Bautin's famous result. |
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