  The stable clustering hypothesis is a key analytical anchor on the nonlinear
dynamics of gravitational clustering in cosmology. It states that on
sufficiently small scales the mean pair velocity approaches zero, or
equivalently, that the mean number of neighbours of a particle remains constant
in time at a given physical separation. In this paper we use N-body simulations
of scale free spectra P(k) \propto k^n with -2 \leq n \leq 0 and of the CDM
spectrum to test for stable clustering using the time evolution and shape of
the correlation function \xi(x,t), and the mean pair velocity on small scales.
For all spectra the results are consistent with the stable clustering
predictions on the smallest scales probed, x < 0.07 x_{nl}(t), where x_{nl}(t)
is the correlation length. The measured stable clustering regime corresponds to
a typical range of 200 \lsim \xi \lsim 2000, though spectra with more small
scale power approach the stable clustering asymptote at larger values of \xi.
  We test the amplitude of \xi predicted by the analytical model of Sheth \&
Jain (1996), and find agreement to within 20\% in the stable clustering regime
for nearly all spectra. For the CDM spectrum the nonlinear \xi is accurately
approximated by this model with n \simeq -2 on physical scales \lsim 100-300
h^{-1} kpc for \sigma_8 = 0.5-1, and on smaller scales at earlier times. The
growth of \xi for CDM-like models is discussed in the context of a power law
parameterization often used to describe galaxy clustering at high redshifts.
The growth parameter \epsilon is computed as a function of time and length
scale, and found to be larger than 1 in the moderately nonlinear regime -- thus
the growth of \xi is much faster on scales of interest than is commonly
assumed.
