In this work, we are interested in isolated crossing periodic orbits in planar piecewise polynomial vector fields defined in two zones separated by a straight line. In particular, in the number of limit cycles of small amplitude. They are all nested and surrounding one equilibrium point or a sliding segment. We provide lower bounds for the local cyclicity for planar piecewise polynomial systems, $M_p^c\left(n\right),$ with degrees 2, 3, $4,$ and 5. More concretely, $M_p^c\left(2\right)\ge 13,$ $M_p^c\left(3\right)\ge 26,$ $M_p^c\left(4\right)\ge 40,$ and $M_p^c\left(5\right)\ge 58.$ The computations use parallelization algorithms.

中文翻译：

---

低次平面分段多项式矢量场中的局部循环

在这项工作中，我们对平面分段多项式矢量场中被直线隔开的两个区域中的孤立交叉周期轨道感兴趣。特别是在小幅度的极限循环中。它们全部嵌套并围绕一个平衡点或一个滑动段。我们为平面分段多项式系统的局部循环提供了下界，${\mathrm{中号}}_p^C（ñ），$ 2、3， $4，$ 5.更具体地说， ${\mathrm{中号}}_p^C（2）\ge 13，$ ${\mathrm{中号}}_p^C（3）\ge 26，$ ${\mathrm{中号}}_p^C（4）\ge 40，$ 和 ${\mathrm{中号}}_p^C（5）\ge 58。$ 计算使用并行化算法。