Z <sub> 2 </ Sub> - Change The Parameters Of The Distribution Of Limit Cycles In Planar Polynomial Vector Field Perturbations Control

A Z2 -equivariant perturbed planar Hamiltonian polynomial vector field is considered.By the control of parameter conditions and using the method of detection functions, the facts of there exist 20-23 limit cycles forming different compound eyes can be proved. Determined and exact parameter groups are given. An incorrect result in a reference has been corrected.The paper includes six parts. In the first part, the latest progresses and some results are stated; In second part, some basic knowledge about Zq - equivariant vector field is stated;In the third part, we introduce the dynamics of the unperturbed system (l) =0 ; In the forth part, for the perturbed system (1) ,we compute all 9 detection functions and give allbifurcation parameter values for Hopf bifurcation, all bifurcation parameter values and bifurcation direction values for homoclinic and heteroclinic bifurcations; In the last two parts, we give five different control conditions to obtain the different configurations of limit cycles and the number of limit cycles .The main result in [Chen et al.,2003] will been corrected in the last part.