Small limit cycles bifurcating from fine focus points in cubic order Z2-equivariant vector fields

In this paper, the existence of 12 small limit cycles is proved for cubic order Z2-equivariant vector fields, which bifurcate from fine focus points. This is a new result in the study of the second part of the 16th Hilbert problem. The system under consideration has a saddle point, or a node, or a focus point (including center) at the origin, and two weak focus points which are symmetric about the origin. It has been shown that the system can exhibit 10 and 12 small limit cycles for some special cases. Further studies are given in this paper to consider all possible cases, and prove that such a Z2-equivariant vector field can have maximal 12 small limit cycles. Fourteen or sixteen small limit cycles, as expected before, are not possible.