  The equations for the sliding of a single block driven by an elastic force
show numerically a fast and a slow step in their dynamics when a dimensionless
parameter is very large, a limit pertinent for many applications. An asymptotic
analysis of the solutions explains well the two sharply different steps of the
stick-slip dynamics. The stick (slow) part takes place along a slow manifold in
the phase space. But, in contrast with standard relaxation dynamics (of van der
Pol type), the slow manifold is always formally attracting and the transition
from slow to fast dynamics occurs because the slow dynamics has a finite time
singularity breaking the assumption of slowness. This makes a new kind of
relaxation oscillation. We show that the response of the stick-slip system to
an external noise displays a progressive speed-up before the transition, in
contrast with the well known critical slowing-down observed in the standard
case.
