Estimation The Upper Bound Of The Number Of Zeros Of Abelian Integral For Two Kinds Of Cubic Hamiltonian Systems

The study on Abelian integrals for planar Hamiltonian systems has both important theoretical significance and broad application background. At present,many mathematicians in this fields are concerned about the weakened Hilbert’s16th problem. We investigate the algebraic structures of Abelian integrals for two kinds of cubic Hamiltonian systems by using the methods of qualitative analysis and bifurcation theory,and estimate the upper bound of the number of zeros of Abelian integrals for Hamiltonian systems. The main contents are the followiing.In Chapter1, we first provide a review of basic results and definitions. Then we introduce the research backgrounds,progresses about this topic and main works of this paper.In Chapter2, we give an upper bound of the number of zeros of the Abelian integral for a kind of cubic Hamiltonian systems with one center. It is proved that Hamilton function H(x,y)=x2±x4+y4correspond the minimum upper bound of the number of zeros of the Abelian integral is B(2n+1)=B(2n+2)≤2[n/2]+3[n-1/2]+4.In Chapter3, we give an upper bound of the number of zeros of the Abelian integral for a kind of cubic Hamiltonian systems with two centers. It is proved that Hamilton function H(x,y)=-x2+x4+y4correspond the minimum upper bound of the number of zeros of the Abelian integral is B1(2n+1)≤8[n/2]+9[n-1/2]+10, B1(2n+2)≤11[n/2]+6[n+1/2], B2(2n+1)=B2(2n+2)≤2[n/2]+3[n-1/2]+6. where B1(m), B2(m) are respectively denoted the upper bound of the number of zeros of the Abelian integral I(h)=∮ΓnQ(x, y)dx-P{x, y)dy which in Σ1=(-1/4,0), E2=(0,+∞).