Random maps in physical systems

We show that functions of type $X_n=P[Z^n]$, where $P[t]$ is a periodic function and $Z$ is a generic real number, can produce sequences such that any string of values $X_s,X_{s+1},\mathrm{.}\mathrm{.}\mathrm{.},X_{s+m}$ is deterministically independent of past and future values. There are no correlations between any values of the sequence. We show that this kind of dynamics can be generated using a recently constructed optical device composed of several Mach–Zehnder interferometers. Quasiperiodic signals can be transformed into random dynamics using nonlinear circuits. We present the results of real experiments with nonlinear circuits that simulate exponential and sine functions.