The Poincaré bifurcation by perturbing a class of cubic Hamiltonian systems

This paper studies the Poincaré bifurcation of the planar vector fields $\dot{x}=H_y(x,y)+\varepsilon f(x,y)$, $\dot{y}=-H_x(x,y)+\varepsilon g(x,y)$, where $0<\left|\varepsilon \right|\ll 1$, $H(x,y)=\alpha x^2+\beta y^2+a{x}^4+b{x}^2{y}^2+c{y}^4,(\alpha ,\beta ,a,b,c)\in R^5,\alpha \beta <0$with $a^2+b^2+c^2\ne 0$, and $f(x,y)$ and $g(x,y)$ are polynomials in $(x,y)$ of the degree $n$. The phase portraits of the unperturbed systems with at least one center can be divided into 10 classes by their phase portraits. For general $n$, we obtain the upper bound of the number of limit cycles bifurcating from period annuli if the first order Melnikov function is not identically zero. The results are new and some of the results in the literatures are improved.