In this paper we consider a class of polynomial planar system with two small parameters, ε and λ, satisfying 0 < ε ≪ λ ≪ 1. The corresponding first order Melnikov function M₁ with respect to ε depends on λ so that it has an expansion of the form \({M_1}(h,\lambda ) = \sum\limits_{k = 0}^\infty {{M_{1k}}(h){\lambda ^k}} \). Assume that M_1k′ (h) is the first non-zero coefficient in the expansion. Then by estimating the number of zeros of M_1k′ (h), we give a lower bound of the maximal number of limit cycles emerging from the period annulus of the unperturbed system for 0 < ε ≪ λ ≪ 1, when k′ = 0 or 1. In addition, for each k ∈ ℕ, an upper bound of the maximal number of zeros of M_1k(h), taking into account their multiplicities, is presented.

在本文中，我们考虑一类具有两个小参数ε和λ的多项式平面系统，满足 0 < ε ≪ λ ≪ 1。对应于ε 的一阶 Melnikov 函数M ₁依赖于λ，因此它具有展开式形式\({M_1}(h,\lambda ) = \sum\limits_{k = 0}^\infty {{M_{1k}}(h){\lambda ^k}} \)。假设M _{1 k'} ( h ) 是展开式中的第一个非零系数。然后通过估计M _{1 k′} ( h)，当k′ = 0 或 1时，我们给出了从无扰系统的周期环中出现的最大极限环数的下限，当k′ = 0 或 1 时，有一个0 < ε ≪ λ ≪ 1。此外，对于每个k ∈ ℕ，一个考虑到它们的多重性，给出了M _{1 k} ( h )的最大零点数的上限。