A classic result in chaos theory is that small perturbations in "chaotic" iterated maps grow roughly exponentially. Notice that once the perturbations have magnitudes of order 1, there is no longer the same kind of growth, and instead there are often seemingly random fluctuations, more characteristic of intrinsic randomness generation.
Many details of arithmetic matter to the results of this Demonstration. The perturbations here are always powers of 10. The arithmetic computations are done to arbitrary precision in Mathematica. Fixed precision arithmetic would give incorrect results.