The complex center and complex isochronous center problems for a complex quartic polynomial system

In this paper, we first study the complex center and complex isochronous center problems for a complex quartic polynomial differential system. More precisely, by calculating and decomposing the variety of the ideal generated by the singular point quantities (resp. complex period quantities), we obtain the necessary conditions for the resonant elementary equilibrium (resp. complex center) of the system to be a complex center (resp. a complex isochronous center). Using time-reversibility and the Darboux integrable theory, we rigorously prove that these conditions are also sufficient. Furthermore, when the coefficients and the variables of the system are complex conjugates, we not only derive the necessary and sufficient conditions for the center-type equilibrium to be both a center and an isochronous center, but we also give the parametric conditions under which the center-type equilibrium of the system becomes a weak focus of order 10. In this case, although the independence of the focal quantities is not satisfied, we still prove that there exist parametric conditions under which 10 limit cycles can bifurcate from this weak focus.