Dynamical Systems describing matter dominating era in Modified Tele-Parallel Gravity

We investigate in the present paper, cosmological dynamical system describing matter evolution era under modified teleparallel theory of gravity. After reconstructing general cosmological dynamical system depending on the parameter $m$ which is function of the Hubble parameter, we explore three different cases of matter dominated era. First, we start with an antaz asymptotic value $m=-9/2$ characterizing matter domination era. Secondly, the present work reveals two other new asymptotic values of the parameter $m$, namely $m=-2$ and $m=-1/2$ corresponding to stiff matter evolution era at early time and dust matter evolution era at late time, respectively. This approach is based on Zel’dovich proposal according to what universe may be filled by stiff matter after inflation and before the radiation epoch. Under analytical and numerical analysis of the reconstructed autonomous dynamical system, the obtained stable fixed point of each evolution era, through its coordinates, leads to ${\omega }_{eff}<-1$ corresponding to phantom-like evolution with dark energy as candidate. This result confirms those existing in literature and suitably cited in this work. Furthermore, matter evolution era can be reached when the dynamical system is rendered non-autonomous. Our approach permits to challenge the reconstruction of $f\left(T\right)$ models describing dark energy when scalar factor of stiff and dust matter is applied.