  This paper is concerned with the following problem: Given a stochastic
non-linear system controlled over a noisy channel, what is the largest class of
channels for which there exist coding and control policies so that the closed
loop system is stochastically stable? Stochastic stability notions considered
are stationarity, ergodicity or asymptotic mean stationarity. We do not
restrict the state space to be compact, for example systems considered can be
driven by unbounded noise. Necessary and sufficient conditions are obtained for
a large class of systems and channels. A generalization of Bode's Integral
Formula for a large class of non-linear systems and information channels is
obtained. The findings generalize existing results for linear systems.
