Conditions for the origin to be a center and the bifurcation of limit cycles in a class of cubic systems (Q1373825)

This paper considers the following cubic system of center and focus type \[ \begin{aligned} \dot x & =y+ax^iy^j =X(x,y) \\ \dot y & =- x+bx^2 +cy^2 +dx^3 +exy^2= Y(x,y) \end{aligned} \] in different cases of \(i\) and \(j\), where \(i+j=2,3\). The authors use the classical method of Lyapunov and calculate focal values using the computer to decide the conditions for the origin to be a center and the order of the origin as a finite focus.