In the qualitative theory of differential equations in the plane one of the most difficult objects to study is the existence of limit cycles. There are many papers dedicated to this subject. Here we will present a survey mainly dedicated to the algebraic and explicit non-algebraic limit cycles of the polynomial differential systems in ${\mathbb{R}}^2$ and of the discontinuous piecewise differential systems in ${\mathbb{R}}^2$ formed by two linear differential systems separated by a straight line. For this class of discontinuous piecewise differential systems the study of their algebraic and explicit non-algebraic limit cycles just is starting. Here we provide the first explicit non-algebraic limit cycle for the discontinuous piecewise linear differential systems. Additionally we recall seven open questions related with these types of limit cycles.

在平面上的微分方程定性理论中，最难研究的对象之一是极限环的存在。有很多关于这个主题的论文。在这里，我们将提出一个主要针对多项式微分系统的代数和显式非代数极限环的调查。${\mathbb{[R}}^{\mathrm{2个}}$ 和中的不连续分段微分系统 ${\mathbb{[R}}^{\mathrm{2个}}$由两个直线分隔的线性微分系统组成。对于这类不连续的分段微分系统，它们的代数和显式非代数极限环的研究才刚刚开始。在这里，我们为不连续的分段线性微分系统提供了第一个明确的非代数极限环。此外，我们还回顾了与这些极限周期类型有关的七个开放性问题。