  We simulate the evolution of one-dimensional gravitating collisionless
systems from non- equilibrium initial conditions, similar to the conditions
that lead to the formation of dark- matter halos in three dimensions. As in the
case of 3D halo formation we find that initially cold, nearly homogeneous
particle distributions collapse to approach a final equilibrium state with a
universal density profile. At small radii, this attractor exhibits a power-law
behavior in density, {\rho}(x) \propto |x|^(-{\gamma}_crit), {\gamma}_crit
\simeq 0.47, slightly but significantly shallower than the value {\gamma} = 1/2
suggested previously. This state develops from the initial conditions through a
process of phase mixing and violent relaxation. This process preserves the
energy ranks of particles. By warming the initial conditions, we illustrate a
cross-over from this power-law final state to a final state containing a
homogeneous core. We further show that inhomogeneous but cold power-law initial
conditions, with initial exponent {\gamma}_i > {\gamma}_crit, do not evolve
toward the attractor but reach a final state that retains their original
power-law behavior in the interior of the profile, indicating a bifurcation in
the final state as a function of the initial exponent. Our results rely on a
high-fidelity event-driven simulation technique.
