Poincare bifurcation of a kind of hamiltonian system under polynomial perturbation

It was proved that the sharp upper bound of the number of ze-ros of Abelian integrals and the number of limit cycles bifurcated from Poincare bifurcation were B(n). An explicit B(n) is derived for the num-ber of zeros of Abelian integrals I(h) = H _Γ(h) f(x; y) dy-g(x; y) dx on the open interval (0;∞), where Γ(h) is an oval lying on the algebraic curve H(x; y) = x2a=A + y2b=B = h, f(x; y),g(x; y) are polynomials of x and y, and max{deg f(x; y); deg g(x; y)} = n. Assume I(h) not vanish iden-tically, c = gcd(a; b), λ = max{a=c; b=c}, then B(n) = 1 2 [n-1 2]([n-1 2]+3) for n ≤ 2λ, B(n) = λ[n-1 2] - 1 2 (λ - 1)(λ - 2) for n ≥ 2λ + 1.