CLASSIFICATION OF THE PHASE PORTRAIT FOR PLANAR Z_8-EQUIVARIANT HAMILTONIAN SYSTEM OF DEGREE 7

In this paper, we consider the Z8-equivariant planar Hamiltonian vector field of degree 7. By using the qualitative and numerical computation, we divide the parameters space into six-parameter-space. And we obtain the results as following : 1. There are seven cases of the number of fixed point of above vector field in finite part, that is, 1,9,l7,25,4l,49, respectively. 2. The possible phase portraits of this vector field are fifty.