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Limit Cycle Bifurcations of a Planar Near-Integrable System with Two Small Parameters
Acta Mathematica Scientia ( IF 1.2 ) Pub Date : 2021-06-01 , DOI: 10.1007/s10473-021-0402-z
Feng Liang , Maoan Han , Chaoyuan Jiang

In this paper we consider a class of polynomial planar system with two small parameters, ε and λ, satisfying 0 < ελ ≪ 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 M1k′ (h) is the first non-zero coefficient in the expansion. Then by estimating the number of zeros of M1k′ (h), we give a lower bound of the maximal number of limit cycles emerging from the period annulus of the unperturbed system for 0 < ελ ≪ 1, when k′ = 0 or 1. In addition, for each k ∈ ℕ, an upper bound of the maximal number of zeros of M1k(h), taking into account their multiplicities, is presented.



中文翻译:

具有两个小参数的平面近似可积系统的极限环分岔

在本文中,我们考虑一类具有两个小参数ελ的多项式平面系统,满足 0 < ελ ≪ 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 < ελ ≪ 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。

更新日期:2021-06-01
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