Abstract
In this paper, Melnikov functions which appear in the study of limit cycles of a perturbed planar Hamiltonian system are studied. There are two main contributions here. The first one is related to the explicit formulae for these functions: a new method is developed to achieve the goal for the second one (Theorem A). the authors also discover a close relation between Melnikov functions and focal quantities (Theorem B). This relation is useful in both judging when a Melnikov function is identically zero and simplifying the computation of a Melnikov function (see §5). Despite these results, this paper also includes other related results, e.g. some estimations of the upper bound for the number of limit cycles in a perturbed Hamiltonian system.
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