Qualitative Analysis For Several Classes Of Polynomial Differential Systems

In this dissertation,we mainly discuss the qualitative analysis of several classes of special fourth polynomial differential systems with the first and the third critical singular point, and a class of codimention-two higher order nonlinear vector field.In part two,we discuss the systemFor the inclusion of the quadratic terms in right side of equations,it is very diffcult to study the globle structure.In each of the system (3),(4),and(5),there is only one finite singular point,and it is saddle-node.According to the theory of the singular index,there is no limit cycle in them. After the discussion of all the possible infinite singular points and the unique finite singular point,we draw out the possible phase portraits of them,there are 43 kinds,12 kinds, and 7 kinds.In this part,we mainly use the Poincare Transformation and the roots of equa-Where P(x,y),Q(x,y) are polynomials of degree not less than 5.We study their global structure, local and globle bifurcations.First we reduce the vector field by normal form theory,then the study is equivalent to the belowFor n = 2 or n = 3,Bogdanov,Takens,Carr etc have been studied it's local bifurcation,after that, Wangminshu,Luodingjun,Lijibin,WangXian etc studied global bifurcations for n = 3.For n = 5, Chenfangyue obtained the homoclinic,hyperclinic bifurcation curves by using Picard-Fuchs equation method.Secondly,we study the changing of fixed points and orbit structure with the changing of the parameters of the reduced vector field.We utilize the typical ways to studying local bifurcations.We only discuss the possibly bifurcations and bifurcation curves of the case a = ?l,and draw the orbit structure diagram.When we study it's globle bifurcation,we use a new method.