Evolutionary behavior of Fractional Holographic Dark Energy within f ( T ) Teleparallel gravity

We investigate the cosmological dynamics of Fractional Holographic Dark Energy within [Formula: see text] gravity by employing the dynamical system approach in a spatially flat Friedmann–Robertson–Walker background. By introducing appropriate dimensionless variables, the field equations are reformulated as a closed system, which allows a systematic phase-space analysis. The resulting system admits four critical points, including two saddle points corresponding to radiation and matter-dominated epochs, and two stable points associated with a DE-dominated phase and a de Sitter solution. The radiation- and matter-dominated critical points are found to possess a saddle character in phase space, ensuring their transient nature and enabling the cosmological evolution to naturally progress toward a stable late-time accelerated attractor. The stable critical points describe accelerated expansion with effective equations of state compatible with DE and de Sitter regimes. The cosmological evolution of the model exhibits a transition from a decelerated to an accelerated expansion phase occurring at [Formula: see text] for [Formula: see text] with present values of the deceleration parameter [Formula: see text] effective EoS [Formula: see text], and FHDE EoS [Formula: see text], indicating consistency with the observed late-time cosmic acceleration. Overall, the analysis indicates that [Formula: see text] gravity is capable of reproducing the standard cosmological sequence within a consistent dynamical framework.