Saddle values and integrability conditions of quadratic differential systems (Q1112230)

The first three saddle values of weak saddle of quadratic differential system (QDE) are computed with Poincaré's method. In order to do this, two theorems are given. The Theorem 1 about the perturbation of a system was used to derive the Theorem 2 which gives the formulae of the three saddle values \(R_ 1,R_ 2,R_ 3\). The necessary and sufficient condition of integrability of the given system, whether real or complex, is \(R_ 1=R_ 2=R_ 3=0\). This conclusion was proved by applying the method and results of Dulac that was published in Bull. Soc. Math. 32, 230-252 (1908). The formulae of focal values for the real quadratic differential equations and a new proof of the famous Bautin's result may be derived by the method given in this paper.