  We obtain the non-linear relation between cosmological density and velocity
perturbations by examining their joint dynamics in a two dimensional
density-velocity divergence phase space. We restrict to spatially flat
cosmologies consisting of pressureless matter and non-clustering dark energy
characterized by a constant equation of state $w$. Using the spherical top-hat
model, we derive the coupled equations that govern the joint evolution of the
perturbations and examine the flow generated by this system. In general, the
initial density and velocity are independent, but requiring that the
perturbations vanish at the big bang time sets a relation between the two. This
relation, which we call the `Zel'dovich curve', acts like an attracting
solution for the phase space dynamics and is the desired non-linear extension
of the density-velocity divergence relation. We obtain a fitting formula for
the curve, which is a generalization of the formulae by Bernardeau and Bilicki
& Chodorowski, and find that as in the linear regime, the explicit dependence
on the dark energy parameters stays weak even in the non-linear regime.
Although the result itself is somewhat expected, the new feature of this work
is the interpretation of the relation in the phase space picture and the
generality of the method. Finally, as an observational implication, we examine
the evolution of galaxy cluster profiles using the spherical infall model for
different values of $w$. We demonstrate that using only the density or the
velocity information to constrain $w$ is subject to degeneracies in other
parameters such as $\sigma_8$ but plotting observations onto the joint
density-velocity phase space can help lift this degeneracy.
