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 $H(2)\ge 4$, $H(3)\ge 8$, $H(n)\ge 7$, for $n\ge 4$ even, and $H(n)\ge 9$, for $n\ge 5$ odd. This improves all the previous results for $n\ge 2$. 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.

在本文中，我们有兴趣为最大极限循环数提供较低的估计 $H（ñ）$ 平面分段线性微分系统，其中两个区域被曲线分开 $ÿ=X^{ñ}$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了更高阶的Melnikov分析。特别是，我们获得$H（\mathrm{2个}）\ge 4$， $H（3）\ge 8$， $H（ñ）\ge 7$， 为了 $ñ\ge 4$ 甚至 $H（ñ）\ge 9$， 为了 $ñ\ge 5$奇怪的。这样可以改善之前所有的结果$ñ\ge \mathrm{2个}$。我们的分析主要基于关于具有正精度的切比雪夫系统和梅尔尼科夫理论的最新结果，这些结果将以任何顺序针对一类具有非线性切换流形的非光滑差分系统进行开发。