Bifurcation of limit cycles in quadratic Hamiltonian systems with various degree polynomial perturbations

In this paper, we consider bifurcation of limit cycles in planar quadratic Hamiltonian systems with various degree polynomial perturbations. Attention is focused on the limit cycles which may appear in the vicinity of an isolated center, and up to 20th-degree polynomial perturbations are investigated. Restricted to the first-order Melnikov function, the method of focus value computation is used to determine the maximal number, H2(n), of small-amplitude limit cycles which may exist in the neighborhood of such a center. Besides the existing results H2(2) = 2 and H2(3) = 5, we shall show that H2(n)=43(n+1) for n = 3, 4, … , 20.