Nilpotent center conditions in cubic switching polynomial Liénard systems by higher-order analysis
Nilpotent center conditions in cubic switching polynomial Liénard systems by higher-order analysis
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DOI：
10.1016/j.jde.2023.10.004
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发表时间：
2023-08
影响因子：
2.4
通讯作者：
Ting Chen;Feng Li;P. Yu
中科院分区：
文献类型：
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作者：
Ting Chen;Feng Li;P. Yu
研究了切换多项式系统中与中心条件和极限环分岔有关的两个重要问题。由于切换多项式系统在非初等奇异点处的Lyapunov常数难以计算，因此使用现有的poincar<s:1> Lyapunov方法研究这两个问题是极其困难的。本文提出了一种考虑切换多项式系统中幂零中心问题的高阶poincarr - lyapunov方法，特别关注了三次切换lisamadard系统。在适当的扰动下，导出了在零中心处切换lisamadard系统的显式中心条件。此外，利用Bogdanov-Takens分岔理论，证明了一类切换lisamadard系统在幂零中心周围存在5个极限环，这是该类多项式系统在幂零中心周围的一个新的环性下界。
The aim of this paper is to investigate two important problems related to nilpotent center conditions and bifurcation of limit cycles in switching polynomial systems. Due to the difficulty in calculating the Lyapunov constants of switching polynomial systems at non-elementary singular points, it is extremely difficult to use the existing Poincaré-Lyapunov method to study these two problems. In this paper, we develop a higher-order Poincaré-Lyapunov method to consider the nilpotent center problem in switching polynomial systems, with particular attention focused on cubic switching Liénard systems. With proper perturbations, explicit center conditions are derived for switching Liénard systems at a nilpotent center. Moreover, with Bogdanov-Takens bifurcation theory, the existence of five limit cycles around the nilpotent center is proved for a class of switching Liénard systems, which is a new lower bound of cyclicity for such polynomial systems around a nilpotent center.