If your underlying system is linear and stable, you can pick any arbitrary precision you are interested in and compute all future behaviour to that precision on a digital computer.

Btw, quantum mechanics is both linear and stable--and even deterministic. Admittedly it's a bit of a mystery how the observed chaotic nature of eg Newtonian billard balls emerges from quantum mechanics.

'Stable' in this case means that small perturbations in the input only lead to small perturbations in the output. You can insert your favourite epsilon-delta formalisation of that concept, if you wish.

To get back to the meat of your comment:

You can simulate such a stable system 'lazily'. Ie you simulate it with any given fixed precision at first, and (only) when someone zooms in to have a closer look at a specific part, you increase the precision of the numbers in your simulation. (Thanks to the finite speed of light, you might even get away with only re-simulating that part of your system with higher fidelity. But I'm not quite sure.)

Remember those fractal explorers like Fractint that used to be all the rage: they were digital at heart---obviously---but you could zoom in arbitrarily as if they had infinite continuous precision.