Cosmological singularities and analytical solutions in varying vacuum cosmologies

We investigate the dynamical features of a large family of running vacuum cosmologies for which $\mathrm{\Lambda }$ evolves as a polynomial in the Hubble parameter. Specifically, using the critical point analysis we study the existence and the stability of singular solutions which describe de-Sitter, radiation and matter dominated eras. We find several classes of $\mathrm{\Lambda }(H)$ cosmologies for which new analytical solutions are given in terms of Laurent expansions. Finally, we show that the Milne universe and the $R_h=ct$ model can be seen as perturbations around a specific $\mathrm{\Lambda }(H)$ model, but this model is unstable.