  Galaxy observations and N-body cosmological simulations produce conflicting
dark matter halo density profiles for galaxy central regions. While simulations
suggest a cuspy and universal density profile (UDP) of this region, the
majority of observations favor variable profiles with a core in the center. In
this paper, we investigate the convergency of standard N-body simulations,
especially in the cusp region, following the approach proposed by
\protect{\citep{13}}. We simulate the well known Hernquist model using the SPH
code Gadget-3 and consider the full array of dynamical parameters of the
particles. We find that, although the cuspy profile is stable, all integrals of
motion characterizing individual particles suffer strong unphysical variations
along the whole halo, revealing an effective interaction between the test
bodies. This result casts doubts on the reliability of the velocity
distribution function obtained in the simulations. Moreover, we find unphysical
Fokker-Planck streams of particles in the cusp region. The same streams should
appear in cosmological N-body simulations, being strong enough to change the
shape of the cusp or even to create it. Our analysis, based on the Hernquist
model and the standard SPH code, strongly suggests that the UDPs generally
found by the cosmological N-body simulations may be a consequence of numerical
effects. A much better understanding of the N-body simulation convergency is
necessary before a 'core-cusp problem' can properly be used to question the
validity of the CDM model.
