The Number Of Zeros Of Abelian Integrals For A Kind Of Two-Parameter Quadratic Hamiltonian System Under Polynomial Perturbations

In this paper, basing on the algebraic-geometric meaning, using Picard-Fuchs equation as tool, refering to the methods of E.Horozov and I.D.Iliev's, combining theory of bifurcation with theory of qualitative analysis , through the symbol computative system, Poincare bifurcation of one kind of two-Parameter quadratic Hamiltonian system under polynomial perturbations is investigated.There are five parts in this paper, concerning with the number of zeros of Abelian integrals of following system under polynomial perturbations:In chapter 1, recalling knowledge of bifurcation theory and present results of the infinitesimal Hilbert 16th problem.In chapter 2, analyzing the singular points, studying the conditions of the existence of homoclinic loops and the trajectory links between saddles, the bifurcation diagram and the global phase portraits of system E(α,β) is obtained.In chapter 3, concentrating on the perturbation of Hamiltonian systems with parameters. Since the complexity of Picard-Fuchs equation, the cases in the neighbourhood of (0, 0) on α — β parameter planar is considered, the upper bound of the number of zeros of Abelian integrals of system E(α,β) under quadratic, cubic and n-order polynomial perturbations are obtained respectively.In chapter 4, basing on the fact that in quadratic case the function I(h) can be written as I(h) = ∫ ∫_Intγ(h)^{ax + by + c}dxdy. The number of intersection points of l : ax + by + c = 0 and the centroid curve L : {((x|ˉ)(h),(y|ˉ)(h)) | h ∈ ∑} gives the number of zeros of I(h). Therefore the study of the geometry of the centroid curve L is crucial for the question. In this part, the method of numerical analysis is used to draw the region of centriod's variance.In chapter 5, the number of zeros of Abelian integrals for one kind of integrable but non-Hamiltonian systems is considered.