On the limit cycles of quadratic differential systems. (Q1565999)

Here, the author gives necessary and sufficient conditions for all finite critical points of the quadratic differential system \[ \dot{x}=dy+\delta x+lx^2+mxy+ny^2, \qquad \dot{y}=x(1+ax+by) , \tag{Q} \] to be weak foci, and he solves an open problem stated by Yanqian Ye. Namely, he proves that, for a suitable selection of values of the coefficients, system (Q) has at least one limit cycle surrounding ``each'' of the four finite critical points ``in turn''; moreover, there exists a connected subset of the parameter space of (Q) such that, when the parameters change in the subset, system (Q) always has at least one limit cycle, and the limit cycles can appear around ``each'' of the four finite critical points ``in turn''.