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.