Darboux Polynomial And Topological Phase Portrait Of The Second Order Riccati Polynomial Differential System

An important issue of dynamic systems is how to get the trajectory of a given differential system. Absolutely, the topological structure of a trajectory will be figured out if the differential system is completely solvable. So far a lot of general methods have been created for differential equations and systems of differential equations. However, that Riccati differential system (x=1,y=a(x)y2+b(x)y+c(x)) as well as its equivalent systems is proved by the French mathematician Liouville at1841to be one of the systems which cannot be solved by the elementary integral method. But Riccati equation played an important role in the history. It was used to prove the solution of the Bessel system is not the elementary function [14]. What’s more, It is also widely used in the fluid mechan-ics and the theory of elastic vibration. So it is still one of the important research issues. In this paper, the polynomial Riccati equation, whose coefficients a(x),b(x) and c(x) were polynomial, was mainly studied. Previously, Zoladek has got its algebraic solution under the condition a(x)=1and b(x)=0, more details see the [13]. Besides, Llibre has proved that this Riccati system has no polynomial first integral and got some Darboux polynomials after expanding Zoladek’s condition to c(x)=k(b(x)-ka(x)){k∈C). But there was no result without this condition. And more details see [1].In this paper, attention was focused on the quadratic polynomial Riccati sys-tem And we got the necessary and sufficient conditions with which the system will have its first-order and second-order Darboux polynomial. At the same time, by using the Poincare compactifcation and the blow up technique we characterize thetopological phase portraits of the system.