Study of dynamical systems and large-scale structure

In this study, we employ dynamical systems methods to analyse the large-scale structure by considering two distinct interaction models (linear and non-linear) within the dark sector, associated with a specific dynamical dark energy model inspired by the Veneziano ghost theory in quantum chromodynamics (QCD). In these models, the dark energy density ($\rho_{DE}$) varies with the Hubble parameter ($H$), expressed as $\rho_{DE} = \alpha H + \beta H^2$. After defining the dimensionless parameters, we present autonomous equations that allow us to find the trace $\text{Tr}(J)$ and the determinant $D(J)$. With these solutions, we demonstrate the presence of unstable, saddle, and stable fixed points, corresponding to the radiation-, matter-, and dark-energy-dominated eras, respectively. Our results suggest that these models are theoretically viable for representing the interaction between dark sector fluids.