论文标题
具有非线性开关曲线的平面分段线性矢量场的高级Melnikov分析
Higher order Melnikov analysis for planar piecewise linear vector fields with nonlinear switching curve
论文作者
论文摘要
在本文中，我们有兴趣对平面分段线性差异系统的最大限制周期$ h（n）$提供较低的估计，其两个区域被曲线$ y = x^n $可以拥有，其中$ n $是一个正整数。为此，我们对线性中心的分段线性扰动执行高阶Melnikov分析。特别是，我们获得了$ h（2）\ geq 4，$ $ h（3）\ geq 8，$ $ h（n）\ geq7，$ for $ n \ geq 4 $偶数，以及$ h（n）\ geq 9，$ n \ geq for $ n \ geq 5 $奇数。这改善了$ n \ geq2的所有先前结果。$我们的分析主要基于有关具有积极准确性的Chebyshev系统和Melnikov理论的最新结果，该结果将以任何类别的非线性开关歧管的非平滑差分系统的顺序开发。
In this paper, we are interested in providing lower estimations for the maximum number of limit cycles $H(n)$ that planar piecewise linear differential systems with two zones separated by the curve $y=x^n$ can have, where $n$ is a positive integer. For this, we perform a higher order Melnikov analysis for piecewise linear perturbations of the linear center. In particular, we obtain that $H(2)\geq 4,$ $H(3)\geq 8,$ $H(n)\geq7,$ for $n\geq 4$ even, and $H(n)\geq 9,$ for $n\geq 5$ odd. This improves all the previous results for $n\geq2.$ Our analysis is mainly based on some recent results about Chebyshev systems with positive accuracy and Melnikov theory, which will be developed at any order for a class of nonsmooth differential systems with nonlinear switching manifold.
