Limit Cycles Near a Centre and a Heteroclinic Loop in a Near-Hamiltonian Differential System

This paper studies limit cycle bifurcation of an (n +1)th order generalized Lienard differential system with an elliptic Hamiltonian H (x, y) = 1/2x(2) - 1/2y(2), -1/4x(4), which bifurcates at least [n/2] limit cycles near either a centre or a heteroclinic loop with two hyperbolic saddles for n <= 150, where about two thirds of the limit cycles are large amplitude ones. To achieve this goal, we provide the precise expression of the third coefficient in the asymptotic expansion of the first order Melnikov function of an analytic near-Hamiltonian system near a polycycle with in (>= 2) hyperbolic saddles, without any prescribed condition, whereas all the known results need. As well as we also characterize its higher order coefficients. Here [a] is the integer part function of a real number a.