Are there interesting examples of open dynamical systems, deterministic or otherwise, with time indexed by \mathbb{Q}_{\geq 0} ? Are they all restrictions of continuous-time systems ?
I am trying to find a streamlined way of dealing with time in (double) categorical systems theory, and it would be nice to have examples beyond \mathbb{N} and \mathbb{R}_{\geq 0}.
I’m not sure about \mathbb{Q}_f, because it doesn’t have a good tangent bundle.
If you are looking for interesting clocks, then in discrete time you could consider non-linear clocks, such as DAGs, and in continuous time you could consider periodic clocks.
My ideas for rational-time systems that might be of interest:
-limits (in some sense) of discrete-time deterministic systems, when the time step goes to 0.
-systems induced by nondeterministic continuous-time systems, for instance with stochastic differential equations, but where you only manipulate finitely many instants at once. Say : I have an ODE with random noise on top, I am given the states at times q_1 < q_2 <...< q_n and the values of the noise at times q_n < q_{n+1} < ... < q_m, what is the joint distribution on the state at times q_{m+1} < ... < q_k ?
(If you’re Markov, you probably only need q_n, not q_1, ..., q_{n-1}, but I’m not entirely sure)
For this approximate system, the update maps take finite-dimensional data as input, and return probability distributions in finite dimension.
This seems like a relevant intermediate notion between “discrete-time” and “continuous-time” nondeterminism.
I like these ideas, but I guess I don’t see the point of using rational numbers instead of real numbers here. The whole point of real numbers is that they are the relevant things for limits.
You should look at how Sam Staton does Brownian motion in LazyPPL: Wiener process regression in LazyPPL; it’s a similar idea. I think it’s harder to do this kind of thing for arbitrary SDEs though; the Wiener process has a nice closed form.
My vague reasoning for “why rational numbers” is the following: if each discrete-time system S^{\delta} has a fixed time step \delta, then there should be a morphism of systems between S^{\delta_1} and S^{\delta_2} when {\delta_1} = n\cdot \delta_2 for some integer n \in \mathbb{N}_{\geq 1}.
For instance, given a \delta \in \mathbb{R}_{>0}, the state/input/output spaces for the system S^{\delta} could be (finite-dimensional) spaces of piecewise linear maps defined on [0,1] that are linear on the intervals [i \cdot \delta; (i+1) \cdot \delta], for 0 \leq i < N, and [N \cdot \delta; 1],
where N is the largest integer such that N \cdot \delta <1.
Then, if {\delta_1} = n\cdot \delta_2, there are inclusions from the state/input/output spaces for S^{\delta_1} to those for S^{\delta_2}, which hopefully define morphisms of systems.
EDIT: this example does not work as stated, the last interval creates issues…
In this framework of discrete-time systems, I am not sure how systems with incommensurable time steps would be compared.
Here, I might be thinking of “limit” in terms of algebra/category theory more than topology; not entirely sure.