ABSTRACT In the present study, we consider planar piecewise linear vector fields with two zones separated by the straight line x = 0. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has centre, a real one for y<0 and a virtual one for y>0, and such that the real centre is a global centre. Then, working with a first-order piecewise linear perturbation we obtain piecewise linear differential systems with three crossing limit cycles. Second, we see that a sliding cycle can be detected after a second-order piecewise linear perturbation. Finally, imposing the existence of a sliding limit cycle we prove that only one adittional crossing limit cycle can appear. Furthermore, we also characterize the stability of the higher amplitude limit cycle and of the infinity. The main techniques used in our proofs are the Melnikov method, the Extended Chebyshev systems with positive accuracy, and the Bendixson transformation.

中文翻译：

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在分段线性平面矢量场中同时出现滑动和交叉极限循环

摘要 在本研究中，我们考虑由直线 x = 0 分隔的两个区域的平面分段线性矢量场。我们的目标是研究此类矢量场同时交叉和滑动极限环的存在。首先，我们为这些系统提供了一个规范形式，假设每个线性系统都有中心，y<0 时为实心，y>0 时为虚心，并且实心是全局中心。然后，使用一阶分段线性扰动，我们获得具有三个交叉极限环的分段线性微分系统。其次，我们看到在二阶分段线性扰动之后可以检测到滑动周期。最后，强加滑动极限环的存在，我们证明只能出现一个额外的交叉极限环。此外，我们还表征了较高振幅极限循环和无穷大的稳定性。我们的证明中使用的主要技术是 Melnikov 方法、具有正精度的扩展 Chebyshev 系统和 Bendixson 变换。