Thirteen limit cycles for a class of Hamiltonian systems under seven-order perturbed terms

In this paper we study the existence, number and distribution of limit cycles of the perturbed Hamiltonian system:$\begin{array}{cc}\hfill & x^{\prime }=4y({\mathit{abx}}^2-{\mathit{by}}^2+1)+\epsilon x\left({\mathit{ux}}^n+{\mathit{vy}}^n-b\frac{\beta +1}{\mu +1}{x}^{\mu }{y}^{\beta }-{\mathit{ux}}^2-\lambda \right)\hfill \\ \hfill & y^{\prime }=4x({\mathit{ax}}^2-{\mathit{aby}}^2-1)+\epsilon y({\mathit{ux}}^n+{\mathit{vy}}^n+{\mathit{bx}}^{\mu }{y}^{\beta }-{\mathit{vy}}^2-\lambda )\hfill \end{array}$where μ + β = n, 0 < a < b < 1, 0 < ε ≪ 1, u, v, λ are the real parameters and n = 2k, k an integer positive.Applying the Abelian integral method Blows TR, Perko LM. Bifurcation of limit cycles from centers and separatrix cycles of planar analytic systems. SIAM Rev 1994;36:341–76] in the case n = 6 we find that the system can have at least 13 limit cycles.Numerical explorations allow us to draw the distribution of limit cycles.