.7 Times Z <sub> 2 </ Sub> - Planar Vector Fields Limit Cycle Branch

The paper consists of four parts. In the first part, Hilbert's 16th problem and the studies on it are introduced, and the results of z_q-equivariant vector fields of degree 5(q = 2,3, 5,6)are given. In the second part, the definitions of z_q-equivariant vector fields and the method of detection functions are started, and all theforms of nontrivial z_q-equivariant planar polynomial vector fields of degree 7 are obtained. In the third part, a concrete numerical example of z₂-equivariant perturbed planar Hamiltonian system of degree 7 is constructed, and for the unperturbed vector field having maximal number of centers, its global phase portriats are analyzed (having at least 9 topologically different phase portraits). Then, for a given parameter group, its phase portrait trend is studied. In the fourth part, by using the bifurcation theory of planar dynamical systems and the method of detection functions, the 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 az₂-equivariant planar polynomial vector field of degree 7 has at least 50 limit cycles with z₂-symmetry.So far, this is the best result of the planar vector field of degree 7 about the limit cycle numbers.