Chaos-induced true randomness

We investigate functions of type Xn=P(θzn), where P(t) is a periodic function, θ and z are real parameters. We show that these functions produce truly random sequences. We prove that a class of autonomous dynamical systems, containing nonlinear terms described by periodic functions of the variables, can generate random dynamics. We generalize these results to dynamical systems with nonlinearities in the form of noninvertible functions. Several examples are studied in detail. We discuss how the complexity of the dynamics depends on the kind of nonlinearity. We present real physical systems that can produce random time-series. We report the results of real experiments using nonlinear circuits with noninvertible I–V characteristics. In particular, we show that a Josephson junction coupled to a chaotic circuit can generate unpredictable dynamics.