  We consider a perturbed ordinary differential equation where the perturbation
is only significant when a one-dimensional null recurrent diffusion is close to
zero. We investigate the first order correction to the unperturbed system and
prove a central limit theorem type result, i.e., that the normalized deviation
process converges in law in the space of continuous functions to a limit
process which we identify. We show that this limit process has a component
which only moves when the limit of the null-recurrent fast motion equals zero.
The set of these times forms a zero-measure Cantor set and therefore the
limiting process cannot be described by a standard SDE. We characterize this
process by its infinitesimal generator (with appropriate boundary conditions)
and we also characterize the process as the weak solution of an SDE that
depends on the local time of the fast motion process. We also investigate the
long time behavior of such a system when the unperturbed motion is trivial. In
this case, we show that the long-time limit is constant on a set of full
Lebesgue measure with probability 1, but it has nontrivial drift and diffusion
components that move only when the fast motion equals zero.
