A Class Z <sub> 8 </ Sub> - Change The Bifurcation Of Limit Cycles Of Planar Vector Fields

The paper consists of two parts. In the first part, the definitions of Zq-equivariant vector fields and the method of detection functions are stated. In the second part, a concrete numerical example of Z8-equivariant perturbed planar Hamiltonian system of of degree 7 is constructed, and for the unperturbed vector field having maximal number of centers, its global phase portraits are analyzed. By using the bifurcation theory of planar dynamical systems and the method of detection functions, this paper gives a configuration of limit cycles forming compound eyes. With the help of numerical analysis (using Maple), it is shown that there exist parameter groups such that a polynomial vector field of degree 7 has at least 49 limit cycles with Z8- symmetry.