Finite-time singularities studying through cosmological dynamical systems survey and reconstruction method in modified teleparallel theory of gravity

Cosmological evolution powered by finite-time singularities is addressed under autonomous and non-autonomous dynamical systems in the framework of [Formula: see text] theory of gravity where [Formula: see text] is the scalar torsion. From approach initiated by Oikonomou et al., we investigate how the dynamical system survey can provide significant explanation on cosmological finite-time singularities. A suitable reconstruction method leads to cosmological [Formula: see text] models. Basing on modified Friedmann equations in a vacuum, we extract an autonomous dynamical system whose application to the de Sitter evolution leads to only one equilibrium fixed point that describes asymptotically a universe in accelerated expansion. When attention is paid to finite-time singularities, the reconstructed dynamical system becomes necessarily non-autonomous. Near the Big-Rip singularity, an analytical resolution approach leads to finite values for the dynamic variables, which not only reveals an accelerated expanding universe but also makes it possible to reconstruct the corresponding [Formula: see text] models. In the case of the three other types of singularity, only type IV leads to an analytical solvable problem, and two of the asymptotic fixed point coordinates diverge. Furthermore, a reconstruction scheme leading to implicit and explicit [Formula: see text] models is undertaken in the last part of this work. It is based on a classical approach using the Hubble parameter of finite-time singularities combined with both Friedman and conservation equations. As results, we obtain in the first-hand implicit [Formula: see text] models for all types of singularity and in the second-hand explicit [Formula: see text] models whose stability is undertaken near each type of singularity in three different evolutions: de Sitter evolution, quintessence-like evolution, and the phantom-like evolution. The relative stability of the reconstructed models not only shows that they cannot describe all four types of singularity simultaneously but also proves that the cosmological evolution is not the same for the four types of singularity. These results are similar to those obtained through dynamical approach survey.