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The study addresses matter-coupled modified gravity, particularly $f (T,\mathcal{T})$ gravity, unveiling distinct formalism. The research furtherdiscusses stability analysis, and the dynamical system approach, exploring thedynamics of critical points to understand these models' viability better. Thedynamical system analysis of the cosmological models in $f(T, \mathcal{T})$gravity, where $T$ and $\mathcal{T}$ respectively represent the torsion scalarand trace of the energy-momentum tensor has been investigated. It demonstrateshow first-order autonomous systems can be treated as cosmological equations andanalyzed using standard dynamical system theory techniques. Two forms of thefunction $f(T,\mathcal{T})$ are considered (i) one with the product of traceand higher order torsion scalar and the other (ii) linear combination of lineartrace and squared torsion. By employing this methodology, the research aims touncover the actual behavior of the Universe. The findings emphasize thegraphical representation of these insights, enriching our understanding ofcosmological scenarios.

Cosmological models in $f (T, \mathcal{T})$ gravity and the dynamical system analysis