We investigate explicit functions that can produce truly random numbers. We use the analytical properties of the explicit functions to show that certain class of autonomous dynamical systems can generate random dynamics. This dynamics presents fundamental differences with the known chaotic systems. We present real physical systems that can produce this kind of random time-series. We report the results of real experiments with nonlinear circuits containing direct evidence for this new phenomenon. In particular, we show that a Josephson junction coupled to a chaotic circuit can generate unpredictable dynamics. Some applications are discussed.

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PACS. 05.45.-a -Nonlinear dynamics and nonlinear dynamical systems. PACS. 42.65.Sf -Dynamics of nonlinear optical systems; optical instabilities, optical chaos and complexity, and optical spatio-temporal dynamics. PACS. 05.45.Vx -Communication using chaos.Abstract. -We show that functions of type Xn = 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 Xs, Xs+1, ..., Xs+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. Other papers have shown that even chaotic communication systems can be cracked if the chaos is predictable [7,8].In the present Letter we will show that using the same experimental setup of Ref.[6] with some small modifications and also other physical systems, it is possible to construct random maps that generate completely unpredictable dynamics.S. Ulam and J. von Neumann [9,10] proved that the logistic map X n+1 = 4X n (1 − X n ) can be solved using the explicit function X n = sin 2 [θπ2 n ]. Other chaotic maps are solvable exactly using, e.g., the functions X n = sin 2 [θπk n ], X n = cos[θπk n ], and other functions of type X n = P [k n ], where k is an integer [11][12][13][14]. For instance, X n = sin 2 [θπ3 n ] is the exact