  Gravitational collapse of dark matter overdensities leads to the formation of
dark matter halos which embed galaxies and galaxy clusters. An intriguing
feature of dark matter halos is that their density profiles closely follow a
universal form irrespective of the initial condition or the corresponding
growth history. This represents a class of dynamical systems with emergent
universalities. We propose an "iterative mean-field approach" to compute the
solutions of the gravitational collapse dynamics. This approach iteratively
searches for the evolution of the interaction field $\phi(t)$ -- in this case
the enclosed mass profile $M(r,t)$ -- that is consistent with the dynamics,
thus that $\phi(t)$ is the fix-point of the iterative mapping,
$\mathcal{H}(\phi) = \phi$. The formalism replaces the N-body interactions with
one-body interactions with the coarse-grained interaction field, and thus
shares the spirit of the mean-field theory in statistical physics. This
"iterative mean-field approach" combines the versatility of numerical
simulations and the comprehensiveness of analytical solutions, and is
particularly powerful in searching for and understanding intermediate
asymptotic states in a wide range of dynamical systems where the solutions can
not be obtained through the traditional self-similar analysis.
