具有积分因子的平面多项式系统的极限环个数

The number of limit cycles of a planar polynomial system with integrating factors

应用Melnikov函数法,研究一类具有积分因子的平面多项式系统在2次多项式扰动下极限环的最大个数问题.推导系统的一阶Melnikov函数的表达式,得到系统的一阶Melnikov函数恒为0;然后探讨系统的二阶Melnikov函数,得出该系统最多有5个极限环(计重数).该研究结果对于一般具有积分因子的多项式系统的极限环个数研究有一定应用价值.

The maximum number of limit cycles for a class of planar polynomial systems with integrating factors under the perturbation of quadratic polynomials is studied using the Melnikov function method.The expression of the first-order Melnikov function for the system is deduced,and it is found to be identically zero.Then,the second-order Melnikov function is explored,and it is concluded that the system can have at most five limit cycles(considering multiplicity).This research has certain application value for the study of the number of limit cycles in general polynomial systems with integrating factors.