  The predictability problem for systems with different characteristic time
scales is investigated. It is shown that even in simple chaotic dynamical
systems, the leading Lyapunov exponent is not sufficient to estimate the
predictability time. This fact is due the saturation of the error on the fast
components of the system which therefore do not contribute to the exponential
growth of the error at large errors. It is proposed to adopt a generalization
of the Lyapunov exponent which is based on the natural concept of error growing
time at finite error size. The method is first illustrated on a simple
numerical model obtained by coupling two Lorenz systems with different time
scales. As a more realistic example, this analysis is then applied to a toy
model of Atmospheric circulation recently introduced by Lorenz.
