  We study a family of dynamical systems obtained by coupling a chaotic
(Anosov) map on the two-dimensional torus -- the chaotic system -- with the
identity map on the one-dimensional torus -- the neutral system -- through a
dissipative interaction. We show that the two systems synchronize: the
trajectories evolve toward an attracting invariant manifold, and the full
dynamics is conjugated to its linearization around the invariant manifold. When
the interaction is small, the evolution of the neutral variable, that is the
variable which describes the neutral system, is very close to the identity;
hence the neutral variable appears as a slow variable with respect to variable
which describes the chaotic system, and which is wherefore named the fast
variable. We demonstrate that, seen on a suitably long time scale, the slow
variable effectively follows the solution of a deterministic differential
equation obtained by averaging over the fast variable.
