%0 Journal Article
%T Poincare bifurcation of a kind of Hamiltonian system under polynomial perturbation
%A Yongkang Zhang
%A Cuiping Li
%J International Journal of Mathematical Analysis
%D 2013
%I 
%X It was proved that the sharp upper bound of the number of zeros ofAbelian integrals and the number of limit cycles bifurcated fromPoincare bifurcation were $B(n)$. An explicit $B(n)$ is derived forthe number of zeros of Abelian integrals $I(h)=oint_{Gamma(h)} f(x,y),dy-g(x,y),dx$ on the open interval $(0,infty)$,where $Gamma(h)$ is an oval lying on the algebraic curve$H(x,y)=x^{2a}/A+y^{2b}/B=h$, $f(x,y)$,$g(x,y)$ arepolynomials of $x$ and $y$, and $max{deg f(x,y),deg g(x,y)}=n$.Assume $I(h)$ not vanish identically, $c=gcd(a,b)$, $lambda=max{a/c,b/c}$,then $B(n)=frac{1}{2}[frac{n-1}{2}]([frac{n-1}{2}]+3)$ for $nleq2lambda$,$B(n)=lambda[frac{n-1}{2}]-frac{1}{2}(lambda-1)(lambda-2)$ for $ngeq2lambda+1$.
%K Abelian integrals
%K Hamiltonian system
%K Poincare bifurcation
%U http://www.m-hikari.com/ijma/ijma-2013/ijma-1-4-2013/zhangykIJMA1-4-2013.pdf