Learning about dynamic systems, deterministic or stochastic, one deals either with a trajectory given by $(X_\alpha)_{\alpha \in A}$ where for a discrete-time process $A = \Bbb N$ and for a continuous time process $A = \Bbb R_+$. From what little I know about Markov random fields, the index set $A$ can also $\Bbb R^n$, but for now I would like to just focus attention on direct analogs of time-based processes.
Are there any good mathematical reasons why we do not work with anything but $A = \Bbb N, \Bbb R_+$? For one thing, from our concept of time it seems to follow that $A$ must be a totally ordered set $(1)$. Perhaps, since a lot of things are done over a finite time horizon, we need such subsets of the index set to be compact, so $A$ must be $\sigma$-compact. Yet, seems the time may need to flow unbounded, $A$ itself must be non-compact. I guess for technical reasons to work with limits over such trajectories it is important that $A$ complete. And also likely $A$ must be homogeneous to talk about time translations.
Question is: are there any math models that use time index set $A$ not being $\Bbb N$ or $\Bbb R_+$ and if not, are there any good reasons for that. In particular, is that the case that a totally ordered set $A$ which is non-compact, $\sigma$-compact, complete and homogeneous in its order topology is isomorphic to an open subset of $\Bbb Z$ or $\Bbb R$?
Let me reply to the first question: there are many such models.
For details and many developments I recommend the books:
by Feres for an emphasis on the relation to ergodic theory and actions of noncompact Lie groups;
by Zimmer for an emphasis on the study of lattices and rigidity (also on Margulis' work);
by Katok and Nitica on higher rank Abelian actions and hyperbolicity.
Note that higher rank Abelian action may be given by $\mathbb R^k$ or $\mathbb Z^k$. This diversity of developments and applications answers to your first question and indeed there are many such models.
But to give specific important examples, I would suggest looking at papers:
by Afraimovich and Chow on topological spatial chaos and homoclinic points of higher rank actions (it may come for example from discretizing a partial differential equation or from an infinite lattice of coupled oscillators);
by Afraimovich and Pesin on the hyperbolicity of travelling waves in lattice models (it is natural to consider a $\mathbb Z^2$ action with the two independent directions relating to space and some discrete time).
I regret that it would be too much to detail here since all this requires a lot of preparations.