Two classes of ODE models with switch-like behavior

In cases where the same real-world system can be modeled both by an ODE system $\bD$ and a Boolean system $\bB$ 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 under suitable assumptions about $\bB$ consistency between $\bD$ and $\bB$ will be implied by sufficient separation of time scales in one class of our models while it may fail in the other class. These results appear to point to more general structure properties that favor consistency between ODE and Boolean models.