In this work, we solve the problem of the coexistence of periodic orbits in homogeneous Boolean graph dynamical systems that are induced by a maxterm or a minterm (Boolean) function, with a direct underlying dependency graph. Specifically, we show that periodic orbits of any period can coexist in both kinds of update schedules, parallel and sequential. This result contrasts with the properties of their counterparts over undirected graphs with the same evolution operators, where fixed points cannot coexist with periodic orbits of other different periods. These results complete the study of the periodic structure of homogeneous Boolean graph dynamical systems on maxterm and minterm functions.

中文翻译：

---

有向图上并行和顺序布尔图动力学系统中周期的共存

在这项工作中，我们解决了由maxterm或minterm（布尔）函数和直接底层依赖图引起的齐次布尔图动力学系统中周期轨道的共存问题。具体来说，我们证明了任何周期的周期性轨道都可以在并行和顺序两种更新时间表中共存。这个结果与具有相同演化算子的​​无向图的对应物的特性形成对比，在固定图不能将定点与其他不同周期的周期轨道共存。这些结果完成了关于maxterm和minterm函数的齐次布尔图动力学系统周期结构的研究。