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Bifurcation of a Kind of Piecewise Smooth Generalized Abel Equation via First and Second Order Analyses
International Journal of Bifurcation and Chaos ( IF 1.9 ) Pub Date : 2020-12-30 , DOI: 10.1142/s0218127420502478
Jianfeng Huang 1 , Zhixiang Peng 1
Affiliation  

In this paper, we consider the problem of estimating the number of nontrivial limit cycles for a kind of piecewise trigonometrical smooth generalized Abel equation with the separation line [Formula: see text]. Under the first and second order analyses, we show that the first two order Melnikov functions of the equation share a same structure which can be studied by an ECT-system. Furthermore, let [Formula: see text] be the maximum number of nontrivial limit cycles of the equation bifurcating from the periodic annulus up to [Formula: see text]th order analysis. We prove that [Formula: see text] and [Formula: see text] (resp., [Formula: see text] and [Formula: see text]) when [Formula: see text] is even (resp., odd).

中文翻译:

一类分段光滑广义Abel方程的一阶和二阶分析分岔

在本文中,我们考虑估计一种分段三角光滑广义阿贝尔方程的非平凡极限环数的问题,该方程具有分隔线[公式:见正文]。在一阶和二阶分析下,我们表明方程的前二阶 Melnikov 函数具有相同的结构,可以通过 ECT 系统进行研究。此外,令 [公式:见文本] 为方程从周期性环分叉到 [公式:见文本] 阶分析的最大非平凡极限环数。当[公式:见文本]为偶数(分别为奇数)时,我们证明[公式:见文本]和[公式:见文本](分别,[公式:见文本]和[公式:见文本])。
更新日期:2020-12-30
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