This paper deals with the number of limit cycles for planar piecewise smooth near-Hamiltonian or near-integrable systems with a switching curve. The main task is to establish a so-called first order Melnikov function which plays a crucial role in the study of the number of limit cycles bifurcated from a periodic annulus. We use the function to study Hopf bifurcation when the periodic annulus has an elementary center as its boundary. As applications, using the first order Melnikov function, we consider the number of limit cycles bifurcated from the periodic annulus of a linear center under piecewise linear polynomial perturbations with three kinds of quadratic switching curves. And we obtain three limit cycles for each case.

中文翻译：

---

具有切换曲线的分段光滑近哈密顿系统的极限环分岔

本文讨论了具有切换曲线的平面分段光滑近哈密尔顿或近可积系统的极限环数。主要任务是建立一个所谓的一阶 Melnikov 函数，该函数在研究从周期环分叉的极限环数中起着至关重要的作用。当周期环以初等中心为边界时，我们使用该函数来研究 Hopf 分岔。作为应用，我们使用一阶Melnikov函数，考虑在分段线性多项式扰动下，从线性中心的周期环分叉出的极限环数，具有三种二次切换曲线。对于每种情况，我们得到三个极限环。