Dynamical Systems Analysis of $f(R, L_m)$ Gravity Model

In this article, we examine the dynamical evolution of flat FRW cosmological model in $f(R, L_m)$ gravity theory. We consider the general form of $f(R, L_m)$ defined as $f(R, L_m) = \Lambda + \frac{\alpha}{2} R + \beta L_m^n$, where $\Lambda$, $\alpha$, $\beta$, $n$ are model parameters, with the matter Lagrangian given by $L_m = -p$. We investigate the model through phase plane analysis, actively studying the evolution of cosmological solutions using dynamical systems techniques. To analyse the evolution equations, we introduce suitable transformations of variables and discuss the corresponding solutions by phase-plane analysis. The nature of critical points is analysed and stable attractors are examined for $f(R, L_m)$ gravity cosmological model. We examine the linear and classical stability of the model and discuss it in detail. Further, we investigate the transition stage of the Universe, i.e. from the early decelerating stage to the present accelerating phase of the Universe by evolution of the effective equation of state, $\{r,s\}$ parameters and statefinder diagnostics for the central values of parameters $\alpha, \beta$ and $n$ constrained using MCMC technique with cosmic chronometer data.