Why do we care about clock systems?

In systems theory, we typically describe systems intensionally, by giving rules for how they evolve over time.

There are many paradigms in which such rules can be given.

How can we relate systems from different rule-paradigms?

The answer is deriving extensional system descriptions--a system is the totality of ways that the system could behave [Willems 2007Willems, J. 2007. The Behavioral Approach to Open and Interconnected Systems. IEEE Control Systems 27, 46–99. http://dx.doi.org/10.1109/MCS.2007.906923.].

The mapping from systems to their behaviors should be functorial, and the existence of clock systems makes this functor representable (a behavior of $X$ is a morphism from the clock into $X$) and gives us a lot of properties for free [Myers 2021Myers, D.J. 2021. Double Categories of Open Dynamical Systems (Extended Abstract). Electronic Proceedings in Theoretical Computer Science 333, 154–167. http://dx.doi.org/10.4204/EPTCS.333.11., Myers 2023Myers, D.J. 2023. Categorical Systems Theory. http://davidjaz.com/Papers/DynamicalBook.pdf.].