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:x′=4y(abx2-by2+1)+εxuxn+vyn-bβ+1μ+1xμyβ-ux2-λy′=4x(ax2-aby2-1)+εy(uxn+vyn+bxμyβ-vy2-λ)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.