Category theory isn't the only way to solve this. Arguably a purely continuous dynamical system like differential equations with certain boundary conditions would similarly work. Discrete dynamical systems however, are much better at representing finite, discrete relationships, particularly with recursion. I'm only just learning about these topics, but simple rules lead to modelling indeterminately complex behaviour (Rule 30, logistic map). These can be viewed quite clearly through the lens of category theory as functors and fixed points. However, the gaps between automata theory, discrete (and non-linear) dynamical systems and finally category theory are still very wide at the moment.