Global Structure And Furcation Curves Of Quadratic Differential Systems With Third-order Weak Singularity

We study the two quadratic differential systems, as follows:The original singularity condition of systems (1) is only decided by parameter c. when|c|<1,original is a saddle, when|e|> l,original is a node; Similarly, the original singularity of systems (2) is only decided by parameter c. When|c|≠ 0,original is a focus. Both of the two systems also have a third-order weak sin-gularity except original, and we denoted it by Aq in this article. The alteration of Ao depends on parameters c and u. The coordinates of Ao in both systems are and Ao is a alteration point and it is not fixed at the origin. This is different from other papers. The conditon of Ao is various, include third or-der weak focus. center, third-order weak saddle, superfine saddle and degenerated singularity, which is rely on c and u on c-u plane. We have got the furcaiton graphs of Aq on the plane of c and u. Lu suggests that all furcaiton curves are clear except a curves called C(α, l)= 0 when he studied the problem of the third-order weak focus. But in the paper, all the furcaiton curves of both systems are clear. Those furcaiton curves divided c-u plane into several areas,Ao is a third-order weak focus or third-order weak saddle in open regions,Ao is a center or superfine saddle or degradated singularity with index +1.We have obtained that not only center but al-so the degradated singularity whose index is+l,in addition, the infinite singularity, can branch out of third-order weak focus or third-order weak saddle.In this paper, we use some methods which are not appeared in other papers when somebody study the plane quadratic systems. As shown below:(1) The intersection of several branches of system is called high order singularity. In those points, some coefficients of system satisfy that both molecular and denominator are equal to zero but the limit of coefficients are exist on the high order point. For example, the origin of c-u plane is a high order point, We can put u = kc into the system, then let c equal to zero. So the system of original is translated to the system which is only about k. Then the system about k is corresponded to the limit system. With this method, we can think that the study on original is translated to the study on a line about parameter k. (2)The method of solving the general integral of systems. At first we get some invariant algebraic curves of the system. Then we can find that the general integral of system is about combination of invariant algebraic curves. For example, there are two invariant algebraic curves of system in the paper, called F2{x,y)= 0> F3(x,y)= 0, Suppose the general integral of system have the form like thisφ(x,y)= (F3)(F2)a.Accorrding to φxP+φyQ= 0,we can solve the undetermined coefficient σ=-3/2.So the general integral of system has been solved.The problem of QC U QW3 is all known that said in Jaume Llibre. There are 9 global phase diagrams of quadratic differential systems with at least one center in previous study. According to the study of this paper, we have got some results: (1) The branch curves of system (1) are:c= 0, u= ±1, u=c, u = -3/5,3u2=c2+2, (13c-9c2+2c3 +15u-24cu + 9c3u - 15u2 + 9cu2) = 0,(13c + 9c2 + 2c3 + 15u + 24cu + 9c2u + 15u2 + 9cu2) = 0. The furcation curves of system (2) are: c= 0, u= c,u=3/5c, 3u2= c2 - 2.All branch curves are linear algebraic curves、quadratic algebraic curves、cubic algebraic curves which are easier to study than the previous systems. (2)In this paper, we obtain that some degradated singularity with index + 1 can branch out of a third order weak focus and two plural singularities according to appropriate perturbation. Similarly, it also can branch out of a saddle and two singularities whose index are + 1.(3) Some infinite degradation singularity also can branch out of center or third-order weak focus from infinite to finite plane.(4)By studying the system (1) in this paper, we have find the higher order singularity than center which is connection center and superfine saddle in the branch curves c=±5.(5)We have got 26 global phase diagrams of quadratic differential systems with at least one center which cover the previous results.