The study on cyclicity of a class of cubic systems

In this paper, we consider a class of cubic systems with polynomial perturbation of the degree at most \begin{document}$ n $\end{document} , and estimate the upper bound of the number of isolated zeros of its Abelian integral. Furthermore, we obtain the distributions of limit cycles bifurcated from a \begin{document}$ Z_4 $\end{document} -equivariant system with \begin{document}$ 5 $\end{document} centers.