Simulating chaotic behavior with finite-state machines

The iterates of the logistic map, xₙ₊₁=rxₙ(1-xₙ), exhibit chaotic behavior for many values of r{∈}[3.57,4.0]. However, if we treat x as an m-state discrete variable, as is done in any digital computer calculation, then the iterates of x invariably form short limit cycles of length {} {}m for chaotic values of r. Although this result questions the validity of digital computer simulations of chaos, we nevertheless find that the statistical properties of the continuous equation, such as the invariant probability distribution and the Lyapunov exponent, are preserved in these cycles. We also find that the transition between periodic and chaotic behavior is still well defined in the discretized map.