In cases where the same real-world system can be modeled both by an ODE system $\mathbb{D}$ and a Boolean system $\mathbb{B}$, it is of interest to identify conditions under which the two systems will be consistent, that is, will make qualitatively equivalent predictions. In this note we introduce two broad classes of relatively simple models that provide a convenient framework for studying such questions. In contrast to the widely known class of Glass networks, the right-hand sides of our ODEs are Lipschitz-continuous. We prove that if $\mathbb{B}$ has certain structures, consistency between $\mathbb{D}$ and $\mathbb{B}$ is implied by sufficient separation of timescales in one class of our models. Namely, if the trajectories of $\mathbb{B}$ are “one-stepping” then we prove a strong form of consistency and if $\mathbb{B}$ has a certain monotonicity property then there is a weaker consistency between $\mathbb{D}$ and $\mathbb{B}$. These results appear to point to more general structure properties that favor consistency between ODE and Boolean models.

在同一现实世界系统可以由 ODE 系统建模的情况下 $\mathbb{D}$ 和一个布尔系统 $\mathbb{乙}$，确定两个系统一致的条件是有意义的，也就是说，将进行定性等效的预测。在这篇笔记中，我们介绍了两大类相对简单的模型，它们为研究此类问题提供了一个方便的框架。与广为人知的玻璃网络类别相比，我们的 ODE 的右侧是 Lipschitz 连续的。我们证明如果$\mathbb{乙}$ 具有一定的结构，之间的一致性 $\mathbb{D}$ 和 $\mathbb{乙}$在我们的一类模型中充分分离时间尺度暗示了这一点。也就是说，如果轨迹$\mathbb{乙}$ 是“一步一步”，那么我们证明了一种强形式的一致性，如果 $\mathbb{乙}$ 具有一定的单调性，则两者之间的一致性较弱 $\mathbb{D}$ 和 $\mathbb{乙}$. 这些结果似乎指向更一般的结构属性，有利于 ODE 和布尔模型之间的一致性。