Evolution of density fluctuations in the Lema\^itre--Tolman--Bondi
  metric containing coupled dark energy
  We consider inhomogeneous spherically symmetric models based on the
Lema\^{i}tre-Tolman-Bondi (LTB) metric, assuming as its source an interactive
mixture of ordinary baryonic matter, cold dark matter and dark energy with a
coupling term proportional to the addition of energy densities of both dark
fluids. We reduce Einstein's field equations to a first order 7-dimensional
autonomous dynamical system of evolution equations and algebraic constraints.
We study in detail the evolution of the energy density and spatial curvature
profiles along the phase space by means of two subspace projections: a
three-dimensional projection associated with the solutions of the
Friedman-Lema\^\i tre-Robertson-Walker metric (invariant subspace) and a
four-dimensional projection describing the evolution of the inhomogeneous
fluctuations. We also classify and study the critical points of the system in
comparison with previous work on similar sources, as well as solving
numerically the equations for initial energy density and curvature profiles
that lead to a spherical bounce whose collapsing time we estimate
appropriately.
