On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems

Limit Cycle Bifurcations by Perturbing a Compound Loop with a Cusp and a Nilpotent Saddle

“…Proof. First, by (6), (10) with = 2, (12), (14), (29), and Theorem 2.2 in [4], we directly obtain (27) with 0 ( ), 3 ( ) given by (30) and (31), respectively. Then we study the expansions of 1 and 3 .…”

“…There have been many works on this topic. For the study of general near-Hamiltonian systems, see [2][3][4][5][6][7][8][9][10][11][12]; and especially for the system (2) with the elliptic case, one can see [13][14][15][16][17] and references therein. In [2][3][4], the number of limit cycles of the system (1) near a homoclinic loop with a cusp of order one or two or a nilpotent saddle of order one (for the definition of an order of a cusp or nilpotent saddle, see [5]) was studied.…”

Parameter identification on Abelian integrals to achieve Chebyshev property

“…For case (b), Yang et al [33] fixed α = β = −2 and proved that the corresponding Abelian integral has at most 4 zeros which can be detected by applying the same method used in [23], and in the same article, the exact same result was obtained for case (c) by fixing α = −2 and β = −1. For cases (e) and (f), it has been proved that the corresponding Abelian integrals have at most 4 zeros but only 3 zeros have been reached, see [14,24,27]. It should be noted that all of these results were obtained by applying the Chebyshev criterion [8,19].…”

Perturbation of a Period Annulus with a Unique Two‐Saddle Cycle in Higher Order Hamiltonian

“…Kazemi et al [14] and Sun [15] studied system (6) of the case = 1, = = 1. They proved that 3 ≤ U(7) ≤ 4 corresponding system (6).…”