Qualitative dynamics of interacting vacuum cosmologies
  We present a phase-space analysis of the qualitative dynamics cosmologies
where dark matter exchanges energy with the vacuum component. We find fixed
points corresponding to power-law solutions where the different components
remain a constant fraction of the total energy density and given an existence
condition for any fixed points with nonvanishing energy transfer. For some
interaction models we find novel fixed points in the presence of a third
noninteracting fluid with constant equation of state, such as radiation, where
the interacting vacuum+matter tracks the evolution of the third fluid,
analogous to tracker solutions previously found for self-interacting scalar
fields. We illustrate the phase-plane behavior, determining the equation of
state and stability of the fixed points in the case of a simple linear
interaction model, for interacting vacuum and dark matter, including the
presence of noninteracting radiation. We give approximate solutions for the
equation of state in matter- or vacuum-dominated solutions in the case of small
interaction parameters.
