A general family of planar piecewise linear ODEs with two zones both having a real focus and separated by a straight line is considered. By analyzing the number of zero points of a new function related to the intersection points of the trajectories of the linear subsystems with the separation line, complete results on the existence and number of limit cycles are obtained. In particular, complete parameter regions for the existence of 1–2 limit cycles are provided with two concrete examples, which will be helpful in studying some kinds of discontinuity-induced bifurcations (i.e., DIBs). Based on the main results, it is obtained that the family of planar piecewise linear ODEs with focus–focus dynamics separated by a straight line can have 3 limit cycles if and only if one subsystem has a real focus and the other one has a virtual focus. Moreover, it is showed that five is the least value of the number of parameters needed in canonical forms of such systems.

考虑具有两个区域的平面分段线性ODE的一般系列，这两个区域都具有真实焦点并被直线分隔。通过分析与线性子系统的轨迹与分隔线的交点有关的新函数的零点数量，可以获得关于极限环的存在和数量的完整结果。特别是，提供了两个具体示例来说明存在1-2个极限环的完整参数区域，这将有助于研究某些类型的不连续性引起的分叉（即DIB）。基于主要结果，可以得出，当且仅当一个子系统具有真实焦点而另一个子系统具有虚拟焦点时，具有焦点-焦点动力学特性的平面分段线性ODE族可以具有3个极限周期。 。此外，