Limit Cycle Bifurcations by Perturbing a Hamiltonian System with a 3-Polycycle Having a Cusp of Order One or Two

Abstract

In this paper, we study the asymptotic expansion of the first order Melnikov function near a 3-polycycle connecting a cusp (of order one or two) to two hyperbolic saddles for a near-Hamiltonian system in the plane. The formulas for the first coefficients of the expansion are given as well as the method of bifurcation of limit cycles. Then we use the results to study two Hamiltonian systems with this 3-polycycle and determine the number and distribution of limit cycles that can bifurcate from the perturbed systems. Moreover, a sharp upper bound for the number of limit cycles bifurcated from the whole periodic annulus is found when there is a cusp of order one.