Dynamical analysis and cosmological viability of varying $G$ and
  $\Lambda$ cosmology
  The cosmological viability of varying $G\left( t\right) $ and $\Lambda \left(
t\right) $ cosmology is discussed by determining the cosmological eras provided
by the theory. Such a study is performed with the determination of the critical
points while stability analysis is performed. The application of
Renormalization group in the ADM formalism of General Relativity provides a
modified second-order theory of gravity where varying $G\left( t\right) $ plays
the role of a minimally coupled field, different from that of Scalar-tensor
theories, while $\Lambda\left( t\right) =\Lambda\left( G\left( t\right) \right)
$ is a potential term. We find that the theory provides two de Sitter phases
and a tracking solution. In the presence of matter source, two new critical
points are introduced, where the matter source contributes to the universe. One
of those points describes the $\Lambda$CDM cosmology and in order for the
solution at the point to be cosmologically viable, it has to be unstable.
Moreover, the second point, where matter exists, describes a universe where the
dark energy parameter for the equation of state has a different value from that
of the cosmological constant.
