Limit cycles of normal size of some classes of quadratic systems in the plane (Q2850793)

We consider a quadratic system NEWLINE\[NEWLINE\frac{dx}{dt}=\sum^2_{i+j=0} a_{ij}x^iy^j\equiv P(x,y),\quad\frac{dy}{dt}=\sum^2_{i+j=0}b_{ij} x^iy^j\equiv Q(x,y).NEWLINE\]NEWLINE It is known that a quadratic system can have only limit cycles enclosing a unique singular point, which is a focus, as well as it can have no more than two foci. The following distributions of limit cycles for quadratic systems (1) are known as: (a) \(1,(1,0)\); (b) \(2,(2, 0)\); (c) \(3,(3,0)\); (d) \((1,1)\); (e) \((2,1)\); (f) \((3,1)\). In this very paper, we suggest the method of construction systems (1) with distribution of (e) and (f) limit cycles for the different configurations of singular points. The existence of the exact given number of limit cycles is proved by using the Dulac function. All limit cycles of the given systems can be detected through numerical methods; i.e. the limit cycles have ``a normal size'', Perko's definition.