Qualitative Analysis Of Several Types Of Polynomial Differential Systems

In this paper, we mainly discuss the topological structure of several classes of special fourth polynomial differential systems with the first and the third critical singular point. As in paper [l],Mr Han Yuliang have discussed the system in the first critical caseand the system in the third critical case(Where b,a30 0 in system(l) and (2))Here, we use the tools and method of paper[l],to study the systemwhere, 040, b04R - {0},where, 640, ao4R-{0}, 631,622513, b04ERFor the system (3) and (4), because of their only one finite singular points ,and they are saddle-nodes,we can easily give the result that there is no limit cycle in their phase portrait. Using the roots of equations:f1(u) =u5 - nu4 - mu3 - lu2 - ku - 1 f2(u) =u5 + nu4 + mu3 + lu2 + ku + 1f1(v) =v4 + kv3 + lv2 + (m + l)v + nf2(v) =v4 + kv3 + lv2 + (m - l)u + nWe discussed all possible infinite singular points, and according to this ,we drew out all kinds of phase portraits of them.Moreover,when we discuss the system(3),we must use theorem 7.3 of Pr. zhang zhifen, In the theorem 7.3,when "0, n < m, the singular point 0(0, 0) is a saddle-node, But Pro. Zhang didn't give the answer of how to draw the phase portrait near the saddle-node,in the more accurate conditions. In this paper, we gave the answer.