Bouncing Scenario in the <i>f</i> ( <i>T</i> ) Modified Gravity Model with Dynamical System Analysis

In this work, we investigate nonsingular bouncing cosmology within the framework of quadratic modified teleparallel gravity described by the nonlinear model [Formula: see text], where [Formula: see text] characterizes the quadratic torsion correction beyond the teleparallel equivalent of General Relativity. Starting from the pure tetrad formulation of [Formula: see text] gravity in a spatially flat FLRW background, we derive the modified Friedmann equations and reformulate the cosmological evolution as a two-dimensional autonomous dynamical system using suitable dimensionless variables associated with the scalar field sector. The phase-space analysis reveals the existence of saddle, unstable, and stable cosmological configurations depending on the stability properties of the critical points. In particular, the scalar field dominated solution behaves as a late-time attractor under appropriate conditions on the potential parameter. To establish the realization of a genuine nonsingular bounce, we reconstruct the cosmological dynamics through the regular scale factor, which remains finite and strictly positive during the entire cosmic evolution. The corresponding Hubble parameter satisfies the standard bouncing conditions [Formula: see text], demonstrating a smooth transition from a contracting phase to an expanding epoch without encountering a spacetime singularity. Furthermore, the torsion scalar, effective energy density, and geometrical background remain finite throughout the bounce phase. We additionally analyze the evolution of the deceleration parameter and the effective equation of state parameter and show that the cosmological dynamics temporarily enters a phantom-like regime near the bounce, allowing the effective violation of the null energy condition required for nonsingular bouncing behavior. The reconstructed bouncing solution is shown to be compatible with the qualitative structure of the autonomous dynamical system. In particular, the cosmological trajectory asymptotically approaches the matter-dominated saddle configuration while avoiding unstable stiff-matter solutions. Our results demonstrate that quadratic [Formula: see text] gravity provides a mathematically self-consistent and physically viable framework for describing regular bouncing cosmology through a unified combination of modified teleparallel gravity, dynamical system analysis, and explicit cosmological reconstruction.