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In this study, we analyze the dynamics of the interaction between dark matterand curvature-driven dark energy in viable $f(R)$ gravity models using theframework of dynamical system analysis. We incorporate this interaction byintroducing a source term in their respective continuity equations, given by $Q= \frac{\kappa^2 \alpha}{3H}\tilde{\rho}_{\rm m}\rho_{\rm curv}$, and examinetwo $f(R)$ gravity models that comply with local gravity constraints andcosmological viability criteria. Our findings reveal subtle modifications tofixed points and their stability criteria when compared to the conventionaldynamical analysis of $f(R)$ gravity models without matter-curvatureinteractions as proposed and examined in prior literature. We determine theparameter limits associated with the stability criteria of critical points ofthe dynamical system for both the models. Additionally, the introduction ofinteraction reveals variations in the dynamical aspects of cosmic evolution,contingent on the range of values for the relevant model parameters. Theseresults are consistent with the observed features of cosmic evolution withinspecific limits of the $f(R)$ models parameters and the coupling parameter$\alpha$. We also examine the evolutionary dynamics of the universe in theinteracting scenario, in the context of cosmological and cosmographicparameters. Our analysis demonstrates that the interacting scenariocomprehensively accounts for all the observed phases of the universe'sevolution and also results in a stable late-time cosmic acceleration.Furthermore, we have examined the impact of the coupling parameter on theevolutionary dynamics, specifically in terms of its influence on the evolutionof the matter-to-curvature energy density ratio.

Dynamics of viable $f(R)$ dark energy models in the presence of Curvature-Matter interactions