关于Kukles系统弱问题(英文)

The weak problem for the Kukles system

讨论与Kukles系统相关的弱问题,给出该系统的前四阶分岔函数表达式,证明了当前四阶分岔函数为零时,其后各阶分岔函数均为零,且该系统原点为中心;证明对于充分小的扰动参数,该系统的极限环的最小上界为3,同时给出该系统存在3个极限环的条件.

This study concerned the weak problem for the Kukles system. The expressions of the first four order Melnikov functions of the Kukles system were specified. Results prove that if the first four Melnikov functions Mj （h） = 0 （j = 1,2,3,4）, then Mk （h） =0 for any k ≥ 5 and the origin of the Kukles system is a centre. It is also shown that the lowest upper bound for the number of limit cycles for the perturbed system is equal to 3. The conditions of the realization of the lowest upper bound are given.