Higher Order Melnikov Functions for Studying Limit Cycles of Some Perturbed Elliptic Hamiltonian Vector Fields

Asheghi, Rasoul; Nabavi, Arefeh

doi:10.1007/s12346-018-0284-1

Cited by 4 publications

(1 citation statement)

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“…The problem has been researched for n = 2 by several research groups independently [4][5][6], and to answer this weakened version, various methods have been developed, among them a popular method is based on the Melnikov functions. By computing the higher-order Melnikov functions based on the algorithm of [7,8], researchers have studied the number of limit cycles bifurcated from the above perturbed system (see [9][10][11][12][13][14][15][16][17][18][19][20]). The results in [12] showed that quasi-homogeneous polynomial Hamiltonian systems have a bound on the number of limit cycle bifurcations from the period annulus at any order of Melnikov functions.…”

Section: Introductionmentioning

confidence: 99%

Third Order Melnikov Functions of a Cubic Center under Cubic Perturbations

2022

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In this paper, cubic perturbations of the integral system (1+x)2dH where H=(x2+y2)/2 are considered. Some useful formulae are deduced that can be used to compute the first three Melnikov functions associated with the perturbed system. By employing the properties of the ETC system and the expressions of the Melnikov functions, the existence of exactly six limit cycles is given. Note that there are many cases for the existence of third-order Melnikov functions, and some existence conditions are very complicated—the corresponding Melnikov functions are not presented.

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