Cosmological dynamics of interacting dark matter-dark energy in generalized Rastall gravity

In this work, we investigate late-time interacting cosmologies within the framework of generalized Rastall gravity, where the interaction arises naturally from the non-conservation of the energy-momentum tensor. We formulate the background evolution of the dark sector as an autonomous dynamical system, defining interaction terms $Q_1=\alpha\,\dot{f}$ and $Q_2=-\dot{f}\,(1+\alpha)$, with $\alpha$ a constant parameter and $f$ a time-dependent function. Three interaction cases are studied: $f \propto \rho_m$, $f \propto \rho_{de}$, and $f \propto \rho_m + \rho_{de}$, assuming a constant dark-energy equation of state $w_{de}$. For each scenario, we derive the closed dynamical system in terms of the density parameters $(\Omega_{de}, \Omega_m)$, identify its fixed points, and analyze their stability across the parameter space. In this context, the phase-space exhibits a standard cosmological dynamics: an unstable radiation point, a transient matter saddle, and a stable late-time attractor with accelerated expansion. In addition, we utilize a joint likelihood analysis with Cosmic Chronometers, PantheonPlus, and DESI data to obtain marginalized parameter estimates at the $68\%$ and $95\%$ confidence levels, constraining the parameter space in each interaction model.