This paper mainly aims to investigate the positive periodic solutions for coupled singular Rayleigh systems. In order to establish the coupled structure, the basic framework of graph theory is employed. By means of Lyapunov method, inequality techniques and a classical consequence of Mawhin’s continuation theorem, some sufficient criterion for the positive periodic solutions has been provided. After that, taken the effects of the delays into account and without imposing more conditions, we further study the positive periodic solutions for a kind of coupled singular Rayleigh system with delays. Here not only the structure is more general than the existing works but the conditions imposed are concise. Consequently, compared with the previous results on the singular systems and coupled systems, the results we established are more generalized and some previous ones can been complemented and improved. Finally, the effectiveness of the established results are validated via an numerical example.

中文翻译：

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耦合奇异瑞利系统的正周期解

本文的主要目的是研究耦合奇异瑞利系统的正周期解。为了建立耦合结构，采用了图论的基本框架。利用Lyapunov方法，不等式技术和Mawhin连续定理的经典结果，为正周期解提供了一些充分的判据。此后，在不考虑更多条件的情况下，考虑了时延的影响，我们进一步研究了一类时滞耦合奇异瑞利系统的正周期解。在这里，不仅结构比现有作品更为笼统，而且条件简明。因此，与先前关于奇异系统和耦合系统的结果相比，我们建立的结果更加概括，可以补充和改进以前的结果。最后，通过数值例子验证了所建立结果的有效性。