@j_bertolotti@bleuje But Etienne is correctly concerned that in chaotic dynamics, the error grows exponentially with time, so you would have to stop very quickly! A more relevant result is the shadowing lemma:
"Although a numerically computed chaotic trajectory diverges exponentially from the true trajectory with the same initial coordinates, there exists an errorless trajectory with a slightly different initial condition that stays near ("shadows") the numerically computed one. Therefore, the fractal structure of chaotic trajectories seen in computer maps is real."