This experiment fundamentally questions randomness as it proves you can mimic any random sequence of numbers (i.e. a sequence of head/tails on a coin) using a chaotic system once the proper initial conditions are derived and fed into the chaotic system. The question then is what is random? Here, we load a random sequence of zeros and ones, then take the chaotic recursive system X[i+1] =4 X[i] (1 - X[i]) and find the appropriate initial condition that will produce the same sequence. To view the Mathematica file click here.

It would also be a good exercise to verify data compression applications using this method. Theoretically, the initial conditions may require more accuracy than the generated sequence for an unknown or derived initial condition (Kolmogorov: A sequence is random if the only algorithm for generating the sequence requires as many bits to describe)