In this paper, we investigate limit cycles induced by threshold nonlinearity of piecewise linear (PWL) differential systems, which are node-focus type or node-center type with the focus or the center being virtual or boundary. To get the number and stability of limit cycles, we adopt a new displacement function with a better configuration than usual. For a given parameter subregion, we exhibit the exact number or the minimum number of limit cycles. In particular, sufficient conditions are established ensuring that there are exactly two limit cycles. When the focus is boundary, we not only show that the maximum number is two, but also verify that the exact number is zero, one or two by varying parameter subregions. Finally, the exact number as well as the stability are obtained in different parameter regions for the PWL differential systems of node-center type.

中文翻译：

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节点焦点或节点中心型平面分段线性系统中阈值非线性引起的极限环

在本文中，我们研究了由分段线性（PWL）微分系统的阈值非线性引起的极限环，这些系统是节点焦点类型或节点中心类型，焦点或中心为虚拟或边界。为了获得极限环的数量和稳定性，我们采用了一种新的位移函数，其配置比以往更好。对于给定的参数子区域，我们展示了极限环的确切数量或最小数量。特别是，建立了充分的条件，以确保恰好有两个极限环。当焦点是边界时，我们不仅显示最大数量为二，而且通过改变参数子区域来验证确切的数量为零、一或二。最后，