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The cyclicity of the period annulus of a reversible quadratic system
Proceedings of the Royal Society of Edinburgh Section A: Mathematics ( IF 1.3 ) Pub Date : 2021-02-10 , DOI: 10.1017/prm.2021.2
Changjian Liu 1 , Chengzhi Li 2 , Jaume Llibre 3
Affiliation  

We prove that perturbing the periodic annulus of the reversible quadratic polynomial differential system $\dot x=y+ax^2$, $\dot y=-x$ with a ≠ 0 inside the class of all quadratic polynomial differential systems we can obtain at most two limit cycles, including their multiplicities. Since the first integral of the unperturbed system contains an exponential function, the traditional methods cannot be applied, except in Figuerasa, Tucker and Villadelprat (2013, J. Diff. Equ., 254, 3647–3663) a computer-assisted method was used. In this paper, we provide a method for studying the problem. This is also the first purely mathematical proof of the conjecture formulated by Dumortier and Roussarie (2009, Discrete Contin. Dyn. Syst., 2, 723–781) for q ⩽ 2. The method may be used in other problems.



中文翻译:

可逆二次系统周期环的周期性

我们证明了在所有二次多项式微分系统的类内扰动可逆二次多项式微分系统 $\dot x=y+ax^2$ , $\dot y=-x$ 的周期我们可以得到最多两个极限环,包括它们的多重性。由于未扰动系统的第一个积分包含指数函数,因此无法应用传统方法,除非在 Figuerasa、Tucker 和 Villadelprat (2013, J. Diff. Equ. , 254 , 3647–3663) 中使用了计算机辅助方法. 在本文中,我们提供了一种研究该问题的方法。这也是 Dumortier 和 Roussarie (2009,离散康定。达因。系统。, 2 , 723–781) 对于q ⩽ 2。该方法可用于其他问题。

更新日期:2021-02-10
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