Bouncing Scenario in the $f(T)$ Modified Gravity Model with Dynamical System Analysis

In this work, we study nonsingular bouncing cosmology in quadratic modified teleparallel gravity $f(T)=T+\beta T^{2}$ within a flat FLRW background. Using dimensionless variables, we derive modified Friedmann equations and reformulate the dynamics as a 2D autonomous system. Phase-space analysis identifies saddle, unstable, and stable critical points, with a scalar field-dominated late-time attractor under specific potential conditions. A genuine nonsingular bounce is realized via a regular scale factor, finite and positive throughout, with Hubble parameter satisfying $H(t_b)=0$, $\dot{H}(t_b)>0$, ensuring a smooth transition from contraction to expansion without singularity. The torsion scalar, effective energy density, and geometry remain finite. The deceleration and effective equation of state parameters show a temporary phantom-like regime near the bounce, violating the null energy condition as required. The reconstructed bounce aligns with the dynamical system's qualitative structure, asymptotically approaching a matter-dominated saddle while avoiding unstable stiff-matter solutions. Our results confirm that quadratic $f(T)$ gravity offers a self-consistent and viable framework for regular bouncing cosmology.