It is known that the Melnikov function method is equivalent to the averaging method for studying the number of limit cycles of planar analytic (or $C^{\infty }$) near-Hamiltonian differential systems. In this paper, we study piecewise smooth near-integrable systems and establish the Melnikov function method and the averaging method for finding limit cycles. We also show the equivalence of the two methods even for systems in high dimensional space. Particularly, we obtain the formula of the second order Melnikov function for planar piecewise near-Hamiltonian systems. We finally provide an application example.

中文翻译：

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分段光滑系统周期轨道的分岔方法

众所周知，梅尔尼科夫函数方法等效于研究平面分析极限环数（或 $C^{\infty }$）近哈密顿微分系统。在本文中，我们研究了分段光滑近可积系统，并建立了Melnikov函数法和求极限环的平均法。我们还显示了这两种方法的等效性，即使对于高维空间中的系统也是如此。特别地，我们获得了平面分段近哈密顿系统的二阶梅尔尼科夫函数的公式。我们最后提供一个应用示例。