Bifurcation Of Exptended Quasi-homogeneous Polynomial Differential System With Minimal Weighted Vector(2,1,2)

In this paper,we study the limit cycles and bifurcation of the extended quasi-homogeneous polynomial differential system where x=(x1,x2)T,Q=(Q1,Q2)T,Q1=ax1x2+bx23,Q2=x1+x22,(a,b,c)?R3.[12]has proved that,when c=0 and(a-2)2+8b<0,the sys-tem has a center.In this paper,we prove that(a,b,c)? R3,there is no limit cycles.Besides,we obtain the global portraits of the system by using quasi-homogeneous blow-up,Poincare-Lyapunov compactificaion for the infinite singularities and Darboux polynomial invariant curves and so on.