A Study of Non-Singular Bounce in Myrzakulov-type $f(R,T)$ Gravity with Chaplygin Gas

This study investigates the non-singular bounce within the framework of Myrzakulov-type $f(R,T) = R + \alpha T + \beta T^2$ gravity by adopting a Chaplygin gas equation of state. We employ two methodologies: a reconstruction scheme via a symmetric scale factor ansatz (Model I) and an autonomous dynamical system analysis (Model II). Our results indicate that the quadratic trace parameter $\beta$ acts as a primary physical driver; specifically, for $\beta<0$, the matter-geometry coupling generates sufficient geometric repulsion to effectively violate the Null Energy Condition (NEC) at high densities without the requirement of exotic matter fields. A numerical scan of the $(\beta, \rho_0)$ parameter space indicates a critical density threshold required to initiate the bounce, below which the Universe follows a singular General Relativity trajectory. The models are shown to be physically viable, with the effective equation of state asymptotically approaching a de Sitter attractor ($w_{\text{eff}} \to -1$) and the squared speed of sound remaining within the stability and causality bounds ($0 \le c_s^2 \le 1$). This study shows that the $f(R,T)$ framework provides a stable, classically geometric alternative to the Big Bang singularity, consistent with both early-universe requirements and late-time accelerated expansion.