Limit Cycle Bifurcations Near a Heteroclinic Loop with Two Nilpotent Cusps of General Order

In this paper, we study the expansion of the first order Melnikov function near a heteroclinic loop with two nilpotent cusps of general order. More precisely, the order of the two cusps is m(1) and m(2) respectively, where m(i) >= 1, m(i) is an element of Z(+). For general m(1) and m(2), we give the expansion of the first order Melnikov function and the formulas for the first few coefficients. We further give a general theorem on the number of limit cycles bifurcated from the heteroclinic loop. These results extend the existing results for m(1) = m(2) = 1, m(1) = m(2) = 2 and m(1) = 1, m(2) = 2. As an application, these results are applied to study the number of limit cycles near a heteroclinic loop with two cusps of different order.