A complete classification on the center-focus problem of a generalized cubic Kukles system with a nilpotent singular point (Q6081667)

Consider the planar polynomial system \[ \begin{array}{l} \frac{{dx}}{{dt}}= y(1+a_{11}x+a_{21}x^2+a_{31}x^3), \\ \frac{{dy}}{{dt}} = b_{20}x^2+b_{11}xy+b_{02}y^2+b_{30}x^3+b_{21}x^2y+b_{12}xy^2+b_{03}y^3 \end{array}\tag{1} \] having the origin as nilpotent singular point. The authors derive conditions on the coefficients \(a_{ij},b_{ij}\) such that the origin is a center and study the bifurcation of small amplitude limit cycles. They prove that in the case \(a_{31} \neq 0\) there are 4 center conditions and at least 8 small-amplitude limit cycles can bifurcate from the origin, in case \(a_{31}= 0\) there are 12 center conditions and at least 7 small-amplitude limit cycles can bifurcate from the origin.