Can we have chaotic motion due to the finite precision of our calculations? I understand chaotic motion to mean that very small perturbations in the initial starting condition can lead to very different trajectories in phase space. For this reason, we can never predict the motion accurately, because we can never have 100% accurate initial conditions.
Can we look at the inability to predict the future states in a different way, related to the precision of our calculations (done on a computer)? Are there situations where we may know the initial conditions with 100% accuracy, but still cannot trust any of the predicted motions, because the motion depends on the accuracy of intermediate calculations, which, being done on a computer, are finite and therefore not perfectly precise?
For example, if I needed to calculate a numeric integral as a step towards a final answer, if my integral were computer to 16 floating point vs 32 floating point accuracy, this would correspond to a difference at the sixteenth significant digit, which could then be sufficient to induce wildly different behavior in the subsequent trajectories.
We could imagine a case that no matter how accurate your calculations were, additional accuracy in the calculations would cause the trajectory to diverge chaotically. Is this phenomenon known to exist, and are there examples of it?
 A: The title question is a bit different from the one in the body of the post, so let's look at them separetelly:

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Can we have chaotic motion due to the finite precision of our calculations?

Yes, Lorenz himself described the phenomenon, calling it computational chaos [Lorenz 1989]:

When one seeks approximate solutions of a set of differential equations by stepwise numerical integration, the choice of a time increment $\tau$ [...] may yield chaotic solutions, even when the true solutions approach limit cycles or fixed points.



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cannot trust any of the predicted motions [?]

At least for hyperbolic systems actually yes, you can trust them. What gets you covered is the so-called shadowing theorem, which guarantees that, even though you're indeed not simulating the true trajectory of the initial condition you picked, there's a slightly different initial point whose trajectory remains arbitrarily close to the computer-generated trajectory. Check also this answer.
[Lorenz 1989] Computational chaos-a prelude to computational instability, Physica D 35 (3), 1989, Pages 299-317.
A: Yes, it is entirely possible that round off errors due to finite precision arithmetic can dramatically affect the outcome of computer simulations of non-linear systems. In fact, one of the pioneers of modern chaos theory, Edward Lorenz, was inspired to study chaotic systems when he experienced this issue. Lorenz was running a weather simulation involving non-linear differential equations on an early digital computer. When he tried to reproduce a scenario by entering initial values with three decimal places of precision, he found that the re-run diverged very quickly from the original output. Investigating the cause of this surprising behaviour, which Lorenz later described as the butterfly effect, led to the discovery of the Lorenz attractor.
