Generalized Logarithmic Equation of State in Classical and Loop Quantum Cosmology Dark Energy-Dark Matter Coupled Systems

In this paper we shall study the phase space of a coupled dark energy-dark matter fluids system, in which the dark energy has a generalized logarithmic corrected equation of state. Particularly, the equation of state for the dark energy will contain a logarithmic function of the dark energy density $\rho_d$ and will also have quadratic and Chaplygin gas-like terms, expressed in terms of $\rho_d$. We shall use the dynamical system approach in order to study the cosmological dynamics, and by appropriately choosing the dynamical system variables, we shall construct an autonomous dynamical system. The study will be performed in the context of classical and loop quantum cosmology, and the focus is on finding stable de Sitter attractors. As we demonstrate, in both the classical and loop quantum cosmology cases, there exist stable de Sitter attractors in the phase space, with the loop quantum cosmology case though having a wider range of the free parameter values for which the stable de Sitter attractors may occur. It is emphasized that the use of a generalized dark energy equation of state makes possible the existence of de Sitter attractors, which were absent in the case that a simple logarithmic term constitutes the dark energy equation of state.