Dynamical system approach to running $\Lambda$ cosmological models
  We discussed the dynamics of cosmological models in which the cosmological
constant term is a time dependent function through the scale factor $a(t)$,
Hubble function $H(t)$, Ricci scalar $R(t)$ and scalar field $\phi(t)$. We
considered five classes of models; two non-covariant parametrization of
$\Lambda$: 1) $\Lambda(H)$CDM cosmologies where $H(t)$ is the Hubble parameter,
2) $\Lambda(a)$CDM cosmologies where $a(t)$ is the scale factor, and three
covariant parametrization of $\Lambda$: 3) $\Lambda(R)$CDM cosmologies, where
$R(t)$ is the Ricci scalar, 4) $\Lambda(\phi)$-cosmologies with diffusion, 5)
$\Lambda(X)$-cosmologies, where
$X=\frac{1}{2}g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}\phi$ is a kinetic
part of density of the scalar field. We also considered the case of an emergent
$\Lambda(a)$ relation obtained from the behavior of trajectories in a
neighborhood of an invariant submanifold. In study of dynamics we use dynamical
system methods for investigating how a evolutional scenario can depend on the
choice of special initial conditions. We showed that methods of dynamical
systems offer the possibility of investigation all admissible solutions of a
running $\Lambda$ cosmology for all initial conditions, their stability,
asymptotic states as well as a nature of the evolution in the early universe
(singularity or bounce) and a long term behavior at the large times. We also
formulated an idea of the emergent cosmological term derived directly from an
approximation of exact dynamics. We show that some non-covariant
parametrizations of Lambda term like $\Lambda(a)$, $\Lambda(H)$ give rise to
pathological and nonphysical behaviour of trajectories in the phase space. This
behaviour disappears if the term $\Lambda(a)$ is emergent from the covariant
parametrization.
