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

Abstract

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.