  We apply the formalism of dynamical system analysis to investigate the
evolution of interacting dark energy scenarios at the background and
perturbation levels in a unified way. Since the resulting dynamical system
contains the extra perturbation variable related to the matter overdensity, the
critical points of the background analysis split, corresponding to different
behavior of matter perturbations, and hence to stability properties. From the
combined analysis, we find critical points that describe the non-accelerating
matter-dominated epoch with the correct growth of matter structure, and the
fact that they are saddle provides the natural exit from this phase.
Furthermore, we find stable attractors at late times corresponding to a dark
energy-dominated accelerated solution with constant matter perturbations, as
required by observations. Thus, interacting cosmology can describe the matter
and dark energy epochs correctly, both at the background and perturbation
levels, which reveals the capabilities of the interaction.
