Quadratic Polynomial System Solution To Study

The differential systems’s bifurcation theory is one of the important research field in ordinary differential equations qualitative theory, which mainly study the law about the vector field topology of the global trajectory changed with parameter change. For the bifurcation theory about planar vector field, the study of limitcentral branch has become a hot problem concernd by mathematicians. In the1900international Congress of Mathematicians, D.Hilbert proposed the famous23mathematical problems,the16th problems in which is to study the the solution’s geometric properties of system x=Pn(x,y),y=Qn(x.y), when Pn(x,y) and Qn(x,y) are realpolynomials of degree n. Since the1980s, many mathematicians dedicated to the study of Hilbert16th problem, however, this problem even for the secondary disturbance Hmaliotn system still did not solve.In1977s V.I.Arnold proposed the the weakening Hilbert16problem.which is to determine the number of zeros of the Abel integral. It transforms the question of the number about limit cycles minimum upper bound in Hmaliotn vector fields change to discuss the corresponding Abel integral I (h) the least upper bound of the number of isolated zeros (number of weight) in its tight branchΣ. But because of the difficulty of solving high order equations,estimating the number of the Abel integral’s zeros is a hard work.Therefore, the problem about weak Hilbert16th problem is still one of today’s hot topics.This paper focuses on the study of the above-mentioned problems, the main content can be summarized as follows:On the basis of the existing literature, the first part study the upper bound of the number about the isolation zeros of the system’s Abel integral,which is under arbitrary n polynomial perturbations, with the help of Picard-Fuchs equation method and the Riccati equations method. The system is belong to a class of quadratic reversible systems with dual center. And we obtained the Abel integral number of zeros does not exceed14n+18, when n≥4.In second part we use the the reflection function of the theory to study reflection function structure when quadratic polynomial system is the simple system.and give the sufficient condition for such a structure function for the reflection function, the behavior of the Poincare map when the secondary cycle system and its periodic solution, and gives example to verify the correctness of the conclusions.Periodic solution to solve complex differential equations provide a good starting point, the paper will use the behavior of periodic solution of the equation method and the method of reflective function of two quadratic differential systems, in order to further explore the differential equations behavior.