Phase-space structure and stability analysis in f(R,Lm,T) gravity

This work explores the cosmological dynamics of the Universe within the framework of the modified gravity model [Formula: see text] using the dynamical system approach. By defining two dimensionless variables, an autonomous system is constructed to analyze the model’s expansion behavior. Three critical points are identified and examined their stability, deceleration parameter, and effective EoS. The point [Formula: see text] represents a matter-dominated and decelerated phase that is either unstable or saddle-type, while [Formula: see text] denotes a stable de-Sitter attractor describing the late-time accelerated expansion. The third point, [Formula: see text], corresponds to a general phase allowing transitions between quintessence, phantom and de-Sitter regimes depending on model parameters. The present model gives [Formula: see text] and [Formula: see text], with corresponding transition redshifts [Formula: see text] and [Formula: see text] for the matter-like and radiation-like configurations, respectively, thereby confirming the existence of a stable late-time accelerating attractor. Moreover, the effective EoS attains [Formula: see text] and [Formula: see text] in the two cases, which lies in the quintessence region and supports the present accelerated expansion. The statefinder analysis further shows that point B reproduces the [Formula: see text]CDM model, whereas point C can reduce to it for specific parameter combinations. The statefinder analysis provides [Formula: see text] and [Formula: see text] for the matter-like and radiation-like configurations, respectively, suggesting Chaplygin gas-like and quintessence-like late-time dark energy behavior with mild deviations from the [Formula: see text]CDM fixed point. Overall, the model effectively unifies standard and dynamical dark energy behaviors within a consistent cosmological framework.