The Number of Limit Cycles From a Quartic Center by the Higher-Order Melnikov Functions

Third Order Melnikov Functions of a Cubic Center under Cubic Perturbations

“…We will study the upper bound of limit cycles of the system by using the higher-order Melnikov functions. Although, our system seems similar to the system in [17,18], it needs more-difficult 1-form decompositions than theirs.…”

“…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.…”

“…using the Melnikov function of any order and proved that the upper bound for the number of limit cycles is three. Following the work of [17], Liu [18] studied the number of limit cycles bifurcated from the origin of the following perturbed system…”