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 [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] [Formula: see text]. For general [Formula: see text] and [Formula: see text], 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 [Formula: see text], [Formula: see text] and [Formula: see text] [Formula: see text]. As an application, these results are applied to study the number of limit cycles near a heteroclinic loop with two cusps of different order.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.