Estimation Of The Number Of Zeros Of Abelian Integral Of A Class Of Quadratic Integrable Systems

Finding the lowest upper bound for the number of zeros of Abelian integrals, which is called weakened Hilbert 16th problem, is an important problem in bifurcation theory of ordinary differential equation. It is close related to determinating the number of limit cycles of a perturbated polylomial Hamiltonian system or integrable system on the plane. It is studied in this thesis that the Abelian integrals for quadratic integrable systemwhere∈is a small parameter, A,B and C are real parameters, f(x,y) and g(x,y) are polynomials of x and y with max{deg f(x,y), deg g(x,y)}=n.LetΓ_h be the compact component of H(x,y)=h.This paper consists of five chapters, chapter 1 is an introduction, In chapter 2, it is discussed the normal form of the quadratic integrable system: x=H_y/M, y=-H_x/M, where H(x,y)= (ax+by+c)^kP₂(x,y), P₂(x,y)=(?)a_ijxⁱy^j,a²+b²â‰ 0,k∈Z,M(x,y)=(ax+by+c)^k-1 and itis showed the topological structure graph of the corresponding phase orbit.In the following chapter, it is considered the integrable system (0.1)o with at least one center. A linear upper bound B(n)≤In is derived for the number of the zeros of Abelian integral I(h) = (?) [M(x,y)g(x,y)]dx-[M(x,y)f(x,y)]dy on the open interval∑, where∑is maximal interval ofexistence ofΓ_h,f(x,y) and g(x,y) are polynomials of x and y, n = max{deg f(x,y),deg g(x,y)}, M(x, y) is the corresponding integrating factor.