J. A. González, L. I. Reyes, J. J. Suárez, L. E. Guerrero, and G. Gutiérrez, *Physica D* __178__, 26 (2003)

**From exactly solvable chaotic maps to stochastic dynamics**

For a class of nonlinear chaotic maps, the exact solution can be written as
X_{n} = P(qk^{n}) ,
where P(t) is a periodic
function, q is a real parameter and k is an
integer number. A generalization of these functions:
X_{n} = P(qz^{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.