An Estimate Of The Number Of Zeros Of Abelian Integrals For A Kind Of Cubic Hamilton System

Research on planar Hamiltonian system corresponding to the structure,analyticity and the number of zeros of the Abelian integral has profound theoretical significance and broad application background,is related to the weakened Hilbert16’th problem closely.Aiming at this problem,in this paper,using the method of qualitative theory of ordinary differential equation and bifurcation theory,studying the bifurcation of a cubic hamntonian system which is symmetrical about x,y axis with centre under n-order perturbations(corresponding to H(x,y)=Ax2+By2+Cx4+Ex2y2+Dy4,AB>0,D≠0,C/A2+D/B2=0is all constant).Hamiltonian system was transformed into H(x,y)=x2+y2-x4+ax2y2+y4by coor-dinate transformation,a=EB2/A|D|.Considering the algebraic structure of Abelian integral, discussing the analyticity and the number of zeros.According to the parameter a,it can be divided into two types.When a≥-2,the corresponding hamiltonian system with a cycles of closed orbitrΓh1={(x,y)|H(x,y)=h,h∈∑1△=(0,1/4)}.When a<-2,the corresponding hamiltonian system with one cycles of closed orbit Γh2{(x,y)|H(x,y)=h,h∈∑2(?)(0,-a/a2+4)},centered around the centre O,and two cycles of closed or-bit Γh-={(x,y)|H(x,y)=h,h∈∑3△=(-a/a2+4,1/4),x<0},Γh+={(x,y|H(x,y)=h,h∈∑3,x>0} centered around the centre S1(-(?)2/2,0),S2((?)2/2,0)separately.The main contents of this paper summarized as follows:The first chapter gives some preliminaries related to the content of this paper,and then introduces the main content of research background,research progress and the text of this topic.The second chapter studies the algebraic structure of Abelian integral for the per-turbed systems corresponding in h∈∑1or∑2.I(h)=α(h)I01+β(h)I03+γ(h)I21+δ(h)I23,degα(h)≤[(n-1)/4]△=p,degβ(h)≤[n-3/4]△=q,degγ(h)≤q,degδ(h)≤p-1is all about h the real polynomial,gives the generating element I01,I03,I21,I23which satis-fies the Picard-Fuchs equation,and the second order homogeneous differential equation between I’01and Z’,Z=Z=a-2/6(a2+4)I03+a+2/2(a2+4)I21+1/3I23.The third chapter exends I’01,Z’ to complex plane and then I(h)analytic continua-tion to complex plane by researching the second order homogeneous differential equation. considering the analyticity of h=0,-a/8,-a/a2+4,-1/4,1/4,which may be non analytic point, bedding for the estimate the number of zeros of the Abelian integral for the perturbed systems corresponding in h∈∑1or∑2.The fourth chapter gives the algebraic structure of I(k+1)(h)(0≤k≤q+1) deg αq+2(h)≤p+3q+3,degδq+2(h)≤p+3q+2,G(h)=4/3h(h-1/4)(h+1/4)(h+a/a2+4) is all about h the real polynomial.On the basis of Riccati equation and the generalized Rolle theorem, to calculate the upper bound of zeros of the Abelian integral for the perturbed systems corresponding in h∈∑1or∑2ofσr(a),The fifth chapter studies the algebraic structure of Abelian integral for the per-turbed systems corresponding in h∈∑3. I(h)=α(h)I01+β(h)I03+γ(h)I21+δ(h)I23+α(h)I11+β(h)I13+γ(h)I31+δ(h)I33,deg∝(h)≤p,degβ(h)≤q,degδ(h)≤p-1,degα(h)≤p,degβ(h)≤q,degγ(h)≤q,degδ(h)≤p-1is all about h the real polynomial,gives the generating element I01,I03,I21,I23which satisfies the Picard-Fuchs equation and I11,I13,I31,I33which satisfies the Picard-Fuchs equation.