From exactly solvable chaotic maps to stochastic dynamics

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For a class of nonlinear chaotic maps, the exact solution can be written as X n = P( θk n ), where P( t) is a periodic function, θ is a real parameter and k is an integer number. A generalization of these functions: X n = P( θz n ), where z is a real parameter, can be proved to produce truly random sequences. Using different functions P( t) we can obtain different distributions for the random sequences. Similar results can be obtained with functions of type X n = h[ f( n)], where f( n) is a chaotic function and h( t) is a noninvertible function. We show that a dynamical system consisting of a chaotic map coupled to a map with a noninvertible nonlinearity can generate random dynamics. We present physical systems with this kind of behavior. We report the results of real experiments with nonlinear circuits and Josephson junctions. We show that these dynamical systems can produce a type of complexity that cannot be observed in common chaotic systems. We discuss applications of these phenomena in dynamics-based computation.