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Local bifurcation of limit cycles and integrability of a class of nilpotent systems

“…Moreover, we proved that 13 limit cycles could be bifurcated from its three-order nilpotent critical point. Although this system is very specific, comparing our result with N(7) ≥ 9 in [23], we give a lower bound of cyclicity of three-order nilpotent critical point for seventh-degree nilpotent systems. The result of this article is helpful to the 16th Hilbert problem, it enriches the literatures as to the Hilbert's 16th problem.…”

“…Let N(n) be the maximum possible number of limit cycles bifurcating from nilpotent critical points for analytic vector fields of degree n. It was found that N(3) ≥ 2, N(5) ≥ 5, N(7) ≥ 9 in [23], N(3) ≥ 3, N(5) ≥ 5 in [17], and for Kukles system with six parameters N(3) ≥ 3 in [11]. Recently, Yirong Liu and Jibin Li proved that N(3) ≥ 8 in [24].…”

Local bifurcation of limit cycles and center problem for a class of quintic nilpotent systems

“…To the best of our knowledge, there are essentially three different ways, the normal form theory [6], the Poincaré return map [12] and Lyapunov functions [13], of studying the center-focus problem of nilpotent critical points, see for instance [3,14,15]. On the other hand, the three tools mentioned above have been also used to generate limit cycles from the critical point, see for instance [15][16][17], respectively.…”

“…Let N(n) be the maximum possible number of limit cycles bifurcating from nilpotent critical points for analytic vector fields of degree n. The authors of [16] got N(3) ≥ 2, N(5) ≥ 5, N(7) ≥ 9; The authors of [15] got N(3) ≥ 3, N(5) ≥ 5; For a family of Kukles system with six parameters, the authors of [17] got N(3) ≥ 3. The authors of [21,22] got N(3) ≥ 7 and N(3) ≥ 8, respectively.…”

“…Consider the perturbed system of (2.5) 16) where X(x, y), Y(x, y) are given by (2.6). Clearly, when 0 < |δ| ≪ 1, in a neighborhood of the origin, there exist one elementary node at the origin and two complex critical points of system (2.16) at (x 1 , y 1 ) and (x 2 , y 2 ), where…”