通过扰动退化中心获得的极限循环

Nabil Rezaiki, A. Boulfoul
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引用次数: 0

摘要

本文讨论当我们在一类度数为\(5\)的全同多项式微分系统内对其进行扰动时,从退化中心\[ \dot{x}=-y(3x^2+y^2),\: \dot{y}=x(x^2-y^2),\]分叉出来的极限循环的最大数量。利用二阶平均理论,我们证明在五阶扰动下,围绕退化中心的周期轨道最多会产生五个极限循环。此外,我们还提供了六个例子,它们恰好产生了\(5, 4, 3, 2, 1\) 和\(0\) 极限循环。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Limit cycles obtained by perturbing a degenerate center
This paper deals with the maximum number of limit cycles bifurcating from the degenerate centre \[ \dot{x}=-y(3x^2+y^2),\: \dot{y}=x(x^2-y^2), \] when we perturb it inside a class of all homogeneous polynomial differential systems of degree \(5\). Using averaging theory of second order, we show that, at most, five limit cycles are produced from the periodic orbits surrounding the degenerate centre under quintic perturbation. In addition, we provide six examples that give rise to exactly \(5, 4, 3, 2, 1\) and \(0\) limit cycles.
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